



<span>**Functions and Integrable Systems**</span>[^1][^2]

<span>Xiaoping Xu</span>

<span>Institute of Mathematics, Academy of Mathematics & System Sciences</span>

<span>Chinese Academy of Sciences, Beijing 10080, P.R.China</span>

<span>**Abstract**</span>

<span>Under a certain condition, we find the explicit formulas for the trace functions of certain intertwining operators among $gl(n)$-modules, introduced by Etingof in connection with the solutions of the Calogero-Sutherland model. If $n=2$, the master function of the trace function is exactly the classical Gauss hypergeometric function. When $n>2$, the master functions of the trace functions are a new family of multiple hypergeometric functions, whose differential property and integral representation are dominated by certain polynomials of integral paths connecting pairs of positive integers. Moreover, we define and explicitly find similar trace functions for $sp(2n)$, which give rise to solutions of the Olshanesky-Perelomov model of type C. The master functions of the trace functions for $sp(2n)$ are similar new multiple path hypergeometric functions. Analogous multiple path hypergeometric functions for orthogonal Lie algebras are defined and studied</span>.

Introduction
============

The Calogero-Sutherland model is an exactly solvable quantum many-body system in one-dimension (cf. \[C\], \[S\]) , whose Hamiltonian is given by $$H_{CS}=\sum_{i=1}^n{\partial}_{x_i}^2+K\sum_{1\leq i<j\leq n}\frac{1}{\sinh^2(x_i-x_j)},\eqno(1.1)$$ where $K$ is a constant. The model was used to study long-range interactions of $n$ particles. It has become an important mathematical object partly because its potential is related to the root system of the special linear algebra $sl(n)$. Olshanesky and Perelomov \[OP\] generalized the Calogero-Sutherland model to the system whose Hamiltonian is given by $$\begin{aligned}
\qquad H_{OP}&=&\sum_{i=1}^n{\partial}_{x_i}^2+\sum_{1\leq i<j\leq n}\left(\frac{K_1}
{\sinh^2(x_i-x_j)}+\frac{K_2}{\sinh^2(x_i+x_j)}\right)\\ &
&+\sum_{i=1}^n\left(\frac{K_3}
{\sinh^2x_i}+\frac{K_4}{\sinh^22x_i}\right),\hspace{7cm}(1.2)\end{aligned}$$ where $K_1,K_2,K_3,K_4$ are constants. The potential of the Olshanesky- Perelomov is related to the root systems of all four families of finite-dimensional classical simple Lie algebras. Solving a system is to find eigenfunctions of its Hamiltonians. Etingof \[Ep\] found that the trace functions of certain intertwining operators among $gl(n)$-modules giving rise to eigenfunctions of $H_{CS}$.

The aim of this paper is to find the functional implication of the Etingof trace. In the case of $gl(2)$, the master function of the Etingof trace function is exactly the classical Gauss hypergeometric function. When $n>2$, under a certain condition on weights, we find that the master functions of the Etingof trace functions are a new family of multiple hypergeometric functions, whose differential property and integral representation are dominated by certain polynomials of integral paths connecting pairs of positive integers. Moreover, we define and explicitly find similar trace functions for $sp(2n)$, which give rise to solutions of the Olshanesky-Perelomov model (1.2) of type C ($K_3=0$). The master functions of the trace functions for $sp(2n)$ are similar new multiple path hypergeometric functions. Based on the combinatorial feature of the path hypergeometric functions for $gl(n)$ and $sp(2n)$, we define and study analogous hypergeometric functions for orthogonal Lie algebras. Furthermore, we prove that certain variations of the Weyl functions of classical simple Lie algebras naturally give solutions of the Olshanesky-Perelomov model with different sets of constants.

Below we give more detailed outline of the background and the results of this paper.

The term of "hypergeometric" was first used by Wallis in Oxford as early as 1655 in his work <span>*Arithmetrica Infinitorm*</span> when referring to any series which could be regarded as a generalization of the ordinary geometric series $\sum_{n=0}^{\infty}z^n$. The well-known hypergeometric series of one-variable $_2F_1(a,b;c;z)$ was introduced and studied by Gauss in his thesis presented at Göttingen in 1812. Apell (1880) first systematically studied double hypergeometric functions and defined his famous four functions of types $F_1$-$F_4$. Horn more completely investigated all the double hypergeometric functions of second order, in 1889, 1931 and 1938, by studying corresponding second order hypergeometric partial differential systems with two independent variables. He found additional ten double hypergeometric functions of types $G_1$-$G_3$ and $H_1$-$H_7$, known as the Horn functions. Lauricella (1893) generalized Apell's four double hypergeometric functions to four families of multiple hypergeometric functions of types $F_A^{(n)},\;F_B^{(n)},
 \;F_C^{(n)}$ and $F_D^{(n)}$. Moreover, he conjectured the existence of fourteen complete triple hypergeometric functions of second order. Saran (1954) found the other ten triple hypergeometric functions of types $F_E$-$F_T$. However, Srivastava (1964) discovered three more complete triple hypergeometric functions of second order of types $H_A,\;H_B$ and $H_C$. Saran (1955) gave integrals of Pochhammer type of the functions of types $F_E$-$F_T$. Pandey (1965) transformed the integral representations of $F_F$ and $F_G$ such that two interesting new triple hypergeometric functions of the generalized Horn type were obtained. Exton (1972, 1973) defined and studied twenty-six quadruple hypergeometric functions of types $K_1$-$K_{21}$ and $D_1$-$D_5$, which are generalizations of the Horn functions. Furthermore, he introduced two families of multiple hypergeometric functions in the line of his quadruple functions. The multiple hypergeometric functions we find this this paper are in the stream of the above hypergeometric functions of classical type. We refer \[Eh\] for the details of the hypergeometric functions of classical type and the references therein.

Throughout this paper, all the vector spaces (algebras) are assumed over ${\mathbb}{C}$, the field of complex numbers. Denote by ${\mathbb}{Z}$ the ring of integers and by ${\mathbb}{N}$ the additive semigroup of nonnegative integers. For two integers $m_1,m_2$, we use the notation: $${\overline}{m_1,m_2}=\left\{\begin{array}{ll}\{m_1,m_1+1,...,m_2\}&\mbox{if}\;\;m_1\leq m_2\\
\emptyset & \mbox{if}\;\;m_1> m_2.\end{array}\right.\eqno(1.3)$$ The general Lie algebra $$gl(n)=\mbox{Span}\{E_{i,j}\mid i,j\in{\overline}{1,n}\}\eqno(1.4)$$ with the Lie bracket $$[A,B]=AB-BA\qquad{\mbox{for}}\;\;A,B\in gl(n),\eqno(1.5)$$ where $E_{i,j}$ is the $n\times n$ matrix whose $(i,j)$-entry is 1 and the others are 0. For $\mu\in{\mathbb}{C}$, we denote $$V_\mu=\mbox{Span}\{(x_1x_2\cdots x_n)^\mu x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n}
\mid i_1,i_2,...,i_n\in{\mathbb}{Z}\},\eqno(1.6)$$ where $x_1,x_2,..., x_n$ are $n$ indeterminates. Then $V_\mu$ becomes a $gl(n)$-module with the action $$E_{i,j}=x_i{\partial}_{x_j}-\mu{\delta}_{i,j}\qquad {\mbox{for}}\;\;i,j\in{\overline}{1,n}.\eqno(1.7)$$ The representation of $gl(n)$ in (1.7) is called a <span>*point representation*</span>.

The subspace $$T=\sum_{i=1}^n{\mathbb}{C}E_{i,i}\eqno(1.8)$$ is a toral Cartan subalgebra of the Lie algebra $gl(n)$. A <span>*weight*</span> ${\lambda}$ of $gl(n)$ is a linear function on $T$ determined by $${\lambda}(E_{i,i})={\lambda}_i\qquad {\mbox{for}}\;\;i\in{\overline}{1,n}.\eqno(1.9)$$ Let $M_{{\lambda}}$ be an irreducible Verma module of $gl(n)$ with the highest weight ${\lambda}$ (e.g. cf. \[Hj\]) and let $M_{\lambda}\hat{\otimes}V_\mu$ be the completion of the tensor $M_{\lambda}\otimes
  V_\mu$ with respect to the topology given by weight subspaces. Suppose that $\Phi: M_{\lambda}{\rightarrow}M_{\lambda}\hat{\otimes}V_\mu $ is a $gl(n)$-module homomorphism. View $\Phi$ as a function in $\{x_1,x_2,...,x_n\}$ taking values in $\mbox{End}\:M_{\lambda}$, the space of linear transformations on $M_{\lambda}$. The <span>*Etingof trace function*</span> $$E(z_1,z_2,...,z_n)=(x_1x_2\cdots x_n)^{-\mu}\mbox{tr}_{_{M_{\lambda}}}\:\Phi z_1^{E_{1,1}}
z_2^{E_{2,2}}\cdots z_n^{E_{n,n}}\eqno(1.10)$$ (cf. \[Ep\]). Changing variables $$z_i=e^{2x_i}\qquad{\mbox{for}}\;\;i\in{\overline}{1,n},\eqno(1.11)$$ we have the equation for the Calogero-Sutherland model (1.1): $$\sum_{i=1}^n(z_i{\partial}_{z_i})^2(\Psi)+K\left(\sum_{1\leq i<j\leq n}\frac{z_iz_j}{(z_i-z_j)^2}
\right)\Psi=\nu \Psi\eqno(1.12)$$ and the equation for the Olshanesky-Perelomov model (1.2): $$\begin{aligned}
\qquad& & \sum_{i=1}^n{\partial}_{z_i}^2(\Psi)+[\sum_{1\leq i<j\leq n}
\left(K_1\frac{z_iz_j}{(z_i-z_j)^2}+K_2\frac{z_iz_j}{(z_iz_j-1)^2}\right)\\ & &+\sum_{i=1}^n
\left(K_3\frac{z_i}{(z_i-1)^2}+K_4\frac{z_i^2}{(z_i^2-1)^2}\right)]\Psi=\nu\Psi,\hspace{5.2cm}
(1.13)\end{aligned}$$ Etingof \[Ep\] proved that the function $$\Psi(z_1,z_2,...,z_n)=\frac{E(z_1,z_2,...,z_n)}{\mbox{tr}_{_{M_{-\rho}}}z_1^{E_{1,1}}
z_2^{E_{2,2}}\cdots z_n^{E_{n,n}}}\eqno(1.14)$$ is a solution of (1.12) for suitable constants $K$ and $\nu$, where $$\rho(E_{i,i})=\frac{n+1}{2}-i\qquad{\mbox{for}}\;\;i\in{\overline}{1,n},\eqno(1.15)$$

Denote $$(c)_i=c(c+1)(c+2)\cdots (c+i-1)\qquad {\mbox{for}}\;\;c\in{\mathbb}{C},\;i\in{\mathbb}{N}.\eqno(1.16)$$ The classical Gauss hypergeometric function $$_2F_1(a,b;c; z)=\sum_{m=0}^{\infty}\frac{(a)_m(b)_m}{m!(c)_m}z^m,\qquad\mbox{where}\;\;a,b,c
\in{\mathbb}{C},\;-c\not\in{\mathbb}{N}.\eqno(1.17)$$ In the case $n=2$, we find that $$E(z_1,z_2)=\frac{z_1^{{\lambda}_1+1}z_2^{{\lambda}_2}}{z_1-z_2}\:_2F_1\left(\mu+1,-\mu;{\lambda}_2-{\lambda}_1,
\frac{z_2}{z_2-z_1}\right)\eqno(1.18)$$ up to a scalar multiple. Thus we can view $_2F_1(\tau_1,\tau_2;{\sigma}; z)$ as the master function of $E(z_1,z_2)$.

Let $${\Gamma}_A=\sum_{1\leq j<i\leq n}{\mathbb}{N}{\epsilon}_{i,j}\eqno(1.19)$$ be the torsion-free additive semigroup of rank $n(n-1)/2$ with ${\epsilon}_{i,j}$ as base elements. For ${\alpha}=\sum_{1\leq j<i\leq n}{\alpha}_{i,j}{\epsilon}_{i,j}\in {\Gamma}_A$, we denote $${\alpha}_{_{{\underline}{1}}}={\alpha}_{{\overline}{n}}=0,\;\;{\alpha}_{_{{\underline}{i}}}=\sum_{r=1}^{i-1}{\alpha}_{i,r},\;\;
{\alpha}_{_{{\overline}{j}}}=\sum_{s=j+1}^n{\alpha}_{s,j}\eqno(1.20)$$ Moreover, we define our $(n(n-1)/2)$-variable hypergeometric function of type A by $${\cal X}_A(\tau_1,..,\tau_n;{\vartheta})\{z_{j,k}\}=\sum_{{\beta}\in{\Gamma}_A}\frac{\left[\prod_{s=1}^{n-1}
(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right](\tau_n)_{_{{\beta}_{{\underline}{n}}}}}
{{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}z^{\beta},\eqno(1.21)$$ where $${\beta}!=\prod_{1\leq k<j\leq n}{\beta}_{j,k}!,\qquad z^{\beta}=\prod_{1\leq k<j\leq n}
z_{j,k}^{{\beta}_{j,k}}.\eqno(1.22)$$ Set $$\xi_{r_2,r_1}^A=\prod_{s=r_1}^{r_2-1}\frac{z_{s}}{z_{r_2}-z_s}\qquad
 {\mbox{for}}\;\;1\leq r_1<r_2\leq n.\eqno(1.23)$$ When $n>2$, we assume $$\mu={\lambda}_i-{\lambda}_{i+1}\qquad{\mbox{for}}\;\;i\in{\overline}{1,n-2}.\eqno(1.24)$$ Under the condition (1.24), we find the Etingof trace function $$E(z_1,z_2,...,z_n)=\frac{\prod_{r=1}^nz_r^{{\lambda}_r+n-r}}{\prod_{1\leq k<j\leq n}(z_k-z_j)}
{\cal X}_A(\mu+1,..,\mu+1,-\mu;{\lambda}_n-\mu)\{\xi_{r_2,r_1}^A\}.\eqno(1.25)$$ The differential property and integral representation of ${\cal X}_A$ are dominated by certain polynomials of integral paths connecting pairs of positive integers (cf. Theorems 5.1 and 5.2). Moreover, the system of partial differential equations for ${\cal X}_A$ is a natural multi-variable analogue of the classical hypergeometric equation (e.g. cf. \[AAR\]).

Fernando \[F\] proved that point representations of finite-dimensional simple Lie algebras exist only for $sl(n)$ and $sp(2n)$. Using the similar point representation of $sp(2n)$ as that of $gl(n)$ in (1.7), we define and find analogues of the Etingof trace functions for $sp(2n)$, which give rise to solutions of the equation (1.13) with suitable constants $K_1,K_2,K_4,\nu$ and $K_3=0$. The master functions of the trace functions for $sp(2n)$ are similar to the function ${\cal X}_A$ in (1.21). Furthermore, we prove that certain variations of the Weyl function of classical simple Lie algebras naturally give solutions of the equation (1.13) with different sets of constants. The results may yield more solutions of the equation (1.13) related to the solutions of the systems of partial differential equations for our path hypergeometric functions, as we show for the case $gl(2)$ (cf. Theorem 2.2).

Our path hypergeometric functions resemble the following well-known multiple hypergeometric functions: Apell's first double hypergeometric function (1880) $$F_1(a_1,a_2,a_3:c;z_1,z_2)=\sum_{m_1,m_2=0}^{\infty}\frac{(a_1)_{m_1+m_2}(a_2)_{m_1}
(a_3)_{m_2}}{m_1!m_2!(c)_{m_1+m_2}}z_1^{m_1}z_2^{m_2},\eqno(1.26)$$ Lauricella's second family of hypergeometric functions (1893) $$\begin{aligned}
 \qquad &&F^{(k)}_B(a_1,...,a_k,b_1,...,b_k;c;z_1,...,z_k)
\\ &=&\sum_{m_1,...,m_k=0}^{\infty}\frac{(a_1)_{m_1}\cdots (a_k)_{m_k}(b_1)_{m_1}\cdots
(b_k)_{m_k}}{(c)_{m_1+\cdots m_k}}\frac{z_1^{m-1}\cdots z_k^{m_k}}{m_1!\cdots m_k!}
\hspace{3.4cm}(1.27)\end{aligned}$$ and Horn's eighth double hypergeometric functions: $$H_5(a_1,a_2;c;z_1,z_2)=\sum_{m_1,m_2=0}^{\infty}\frac{(a_1)_{2m_1+m_2}(a_2)_{m_2-m_1}}
{m_1!m_2!(c)_{m_2}}z_1^{m_1}z_2^{m_2}\eqno(1.28)$$ (cf. \[Eh\] for details). Heckman and Opdam \[HO, Hg1,O1, O2, BO\] introduced hypergeometric equations related to root systems and analogous to (1.12). They proved the existence of solutions (hypergeometric functions) of their equations. In our case, the interesting functions are the path hypergeometric functions like ${\cal X}_A$ in (1.21) and the functions like $\Psi$ in (1.14) are not interesting from pure function point of view. Gel'fand and Graev studied analogues of classical hypergeometric functions (so called GG-functions) by generalizing the differential property of the classical hypergeometric functions (e.g. cf. \[GG\]).

The paper is organized as follows. In Section 2, we calculate the Etingof trace function for $gl(2)$ and show a connection between the equation (1.12) with $n=2$ and the classical hypergeometric equation for the Gauss hypergeometric function. This section serves as a contraction of the whole paper. We obtain (1.25) in Section 3. Section 4 is devoted to defining and calculating the analogue of the Etingof trace function for $sp(2n)$. In Section 5, we find the differential property and systems of partial differential equations for the multiple hypergeometric functions of types A and C obtained in Section 3 and Section 4, as the master functions of the Etingof trace functions. The integral representations of our multiple hypergeometric functions of type A are obtained. Analogous multiple hypergeometric functions associated with the root systems of the orthogonal Lie algebras are introduced and studied. In Section 6, we show a connection between the equation (1.13) and Weyl functions of classical Lie algebras. Section 7 is the proof of that the trace functions of certain intertwining operators among $sp(2n)$-modules give rise to solutions of the equation (1.13) with $K_3=0$.

Preliminary
===========

In this section, we find Etingof trace functions for $gl(2)$ in terms of classical Gauss hypergeometric functions. Moreover, we show a connection between the equation (1.12) with $n=2$ and the classical hypergeometric equation.

Let ${\lambda}$ be a weight of $gl(2)$ such that $${\sigma}={\lambda}_1-{\lambda}_2\not\in{\mathbb}{N}\eqno(2.1)$$ (cf. (1.9)). Choose $E_{1,2}$ as the positive root vector. The Verma $gl(2)$-module with the highest-weight vector $v_{\lambda}$ of weight ${\lambda}$ is given by $$M_{\lambda}=\mbox{Span}\{E_{2,1}^iv_{\lambda}\mid i\in{\mathbb}{N}\}\eqno(2.2)$$ with the action $$E_{1,,2}(E_{2,1}^iv_{\lambda})=i({\sigma}+1-i)E_{2,1}^{i-1}v_{\lambda},\qquad E_{2,1}(E_{2,1}^iv_{\lambda})
=E_{2,1}^{i+1}v_{\lambda},\eqno(2.3)$$ $$E_{1,1}(E_{2,1}^iv_{\lambda})=({\lambda}_1-i)E_{2,1}^iv_{\lambda},\qquad E_{2,2}(E_{2,1}^iv_{\lambda})
=({\lambda}_2+i)E_{2,1}^iv_{\lambda}.\eqno(2.4)$$ Under the assumption (2.1), $M_{\lambda}$ is an irreducible $gl(2)$-module.

Let $\mu\in{\mathbb}{C}$ be a fixed constant and let $${\tilde}{M}=\{\sum_{i_1,i_2\in{\mathbb}{Z}}v_{i_1,i_2}x_1^{i_1+\mu}x_2^{i_2+\mu}\mid v_{i_1,i_2}\in
M_{\lambda}\},\eqno(2.5)$$ a space of Laurent series with coefficients in $M_{\lambda}$. Define the action of $gl(2)$ on ${\tilde}{M}$ by $$\begin{aligned}
E_{j_1,j_2}(\sum_{i_1,i_2\in{\mathbb}{Z}}v_{i_1,i_2}x_1^{i_1+\mu}x_2^{i_2+\mu})
&=&\sum_{i_1,i_2\in{\mathbb}{Z}}[E_{j_1,j_2}(v_{i_1,i_2})x_1^{i_1+\mu}x_2^{i_2+\mu}
\\ & &+v_{i_1,i_2}(x_{j_1}{\partial}_{x_{j_2}}-\mu{\delta}_{j_1,j_2})(x_1^{i_1+\mu}x_2^{i_2+\mu})]
\hspace{2.7cm}(2.6)\end{aligned}$$ for $j_1,j_2\in\{1,2\}$. Then ${\tilde}{M}$ becomes a $gl(2)$-module that is isomorphic to $M_{\lambda}\hat{\otimes}V_\mu$ (cf. (1.6)). Recall that a singular vector in a Verma module is a nonzero weight vector annihilated by positive root vectors. Any singular vector of weight ${\lambda}$ in ${\tilde}{M}$ is of the form: $$u_{\lambda}=(x_1x_2)^\mu\sum_{i=0}^{\infty}a_iE_{2,1}^iv_{\lambda}x_1^ix_2^{-i}\eqno(2.7)$$ Observe that $$\begin{aligned}
0&=&E_{1,2}(u_{\lambda})\\ &=&(x_1x_2)^\mu\sum_{i=0}^{\infty}a_i[i({\sigma}+1-i)
E_{2,1}^{i-1}v_{\lambda}x_1^ix_2^{-i}+(\mu-i)E_{2,1}^iv_{\lambda}x_1^{i+1}x_2^{-i-1}]
\\ &=&(x_1x_2)^\mu\sum_{i=0}^{\infty}((\mu-i)a_i+(i+1)({\sigma}-i)a_{i+1})v_{\lambda}x_1^{i+1}x_2^{-i-1}.\hspace{4.1cm}(2.8)\end{aligned}$$ Thus $$(\mu-i)a_i+(i+1)({\sigma}-i)a_{i+1}=0\qquad{\mbox{for}}\;\;i\in{\mathbb}{N},\eqno(2.9)$$ equivalently, $$a_{i+1}=-\frac{(\mu-i)a_i}{(i+1)({\sigma}-i)}\qquad{\mbox{for}}\;\;i\in{\mathbb}{N},\eqno(2.10)$$ For any constant $c$ and nonnegative integer $i$, we denote $${\langle}c{\rangle}_i=c(c-1)\cdots (c-(i-1)).\eqno(2.11)$$ We normalize $u_{\lambda}$ by taking $$a_0=1.\eqno(2.12)$$ By induction, $$a_i=\frac{(-1)^i{\langle}\mu{\rangle}_i}{i!{\langle}{\sigma}{\rangle}_i}.\eqno(2.13)$$ Thus $$u_{\lambda}=\sum_{i=0}^{\infty}\frac{(-1)^i{\langle}\mu{\rangle}_i}{i!{\langle}{\sigma}{\rangle}_i}E_{2,1}^iv_{\lambda}x_1^{i+\mu}x_2^{-i+\mu}.\eqno(2.14)$$

Let $\Phi: M_{\lambda}{\rightarrow}{\tilde}{M}$ be the Lie algebra module homomorphism such that $\Phi(v_{\lambda})=u_{\lambda}$. Note that $$E_{2,1}^j(E_{2,1}^iv_{\lambda}x_1^{i+\mu}x_2^{-i+\mu})=\sum_{k=0}^j\left(\!\!\begin{array}{c}j\\
 k\end{array}\!\!\right){\langle}\mu+i{\rangle}_kE_{2,1}^{i+j-k}v_{\lambda}x_1^{i-k+\mu}x_2^{-i+k+\mu}.
 \eqno(2.15)$$ Moreover, for any $\iota\in{\mathbb}{N}$, we have $$\sum_{k=\iota}^{\infty}\left(\!\!\begin{array}{c}k\\ \iota\end{array}\!\!\right)z^k=
z^{\iota}\sum_{k=\iota}^{\infty}\left(\!\!\begin{array}{c}k\\ \iota\end{array}\!\!\right)
z^{k-\iota}=\frac{z^{\iota}}{\iota !}\frac{d^\iota}{dz^\iota}(\sum_{k=0}^{\infty}z^k)
=\frac{z^{\iota}}{\iota !}\frac{d^\iota}{dz^\iota}\left(\frac{1}{1-z}\right)=\frac{z^{\iota}}
{(1-z)^{\iota+1}}.\eqno(2.16)$$ View $\Phi$ as a "function taking values in $\mbox{End}\:M_{\lambda}$." By (2.15) and (2.16), the Etingof trace function $$\begin{aligned}
& &E(z_1,z_2)=(x_1x_2)^{-\mu}\mbox{tr}_{_{M_{\lambda}}}\:\Phi
z_1^{E_{1,1}}z_2^{E_{2,2}}\\
&=&z_1^{{\lambda}_1}z_2^{{\lambda}_2}\sum_{j=0}^{\infty}\sum_{i=0}^\infty
\left(\!\!\begin{array}{c}j\\
i\end{array}\!\!\right){\langle}\mu+i{\rangle}_i\frac{(-1)^i{\langle}\mu{\rangle}_i}{i!{\langle}{\sigma}{\rangle}_i}\left(\frac{z_2}{z_1}\right)^j\hspace{8cm}\end{aligned}$$ $$\begin{aligned}
 &=&
z_1^{{\lambda}_1}z_2^{{\lambda}_2}\sum_{i=0}^\infty{\langle}\mu+i{\rangle}_i\frac{(-1)^i{\langle}\mu{\rangle}_i}{i!{\langle}{\sigma}{\rangle}_i}
\sum_{j=0}^{\infty}\left(\!\!\begin{array}{c}j\\
i\end{array}\!\!\right)\left(\frac{z_2}{z_1}
\right)^j\\&=&z_1^{{\lambda}_1}z_2^{{\lambda}_2}\sum_{i=0}^\infty{\langle}\mu+i{\rangle}_i\frac{(-1)^i{\langle}\mu{\rangle}_i}{i!{\langle}{\sigma}{\rangle}_i}\left(\frac{z_2}{z_1}\right)^i\left(\frac{1}{1-z_2/z_1}\right)^{i+1}\\
&=&\frac{z_1^{{\lambda}_1+1}z_2^{{\lambda}_2}}{z_1-z_2}\sum_{i=0}^\infty\frac{{\langle}\mu+i{\rangle}_i{\langle}\mu{\rangle}_i}{i!{\langle}{\sigma}{\rangle}_i}\left(\frac{z_2}{z_2-z_1}\right)^i.\hspace{7.2cm}(2.17)\end{aligned}$$

Recall the notion $(c)_n$ in (1.16), which is widely used in the theory of hypergeometric functions. Note $${\langle}c+n{\rangle}_n=(c+1)_n,\;\;{\langle}c{\rangle}_n=(-1)^n(-c)_n\qquad{\mbox{for}}\;\;c\in{\mathbb}{C},\;n\in{\mathbb}{N}.
\eqno(2.18)$$ By (1.17), (2.1), (2.17) and (2.18), we have:

<span>**Theorem 2.1**</span>. <span>*The Etingof trace function for $gl(2)$ is*</span> $$E(z_1,z_2)=\frac{z_1^{{\lambda}_1+1}z_2^{{\lambda}_2}}{z_1-z_2}\:_2F_1\left(\mu+1,-\mu;{\lambda}_2-{\lambda}_1:
\frac{z_2}{z_2-z_1}\right).\eqno(2.19)$$

The function in (1.14) $$\Psi(z_1,z_2)=z_1^{{\lambda}_1+1/2}z_2^{{\lambda}_2-1/2}\:_2F_1\left(\mu+1,-\mu;{\lambda}_2-{\lambda}_1:
\frac{z_2}{z_2-z_1}\right)\eqno(2.20)$$

Next, we will give a connection between the equation (1.12) with $n=2$ and the classical hypergeometric equation. For $\mu_1,\mu_2\in{\mathbb}{C}$, we set $$\phi_{\mu_1,\mu_2}(z_1,z_2)=(z_1z_2)^{\mu_1}(z_1-z_2)^{\mu_2}.\eqno(2.21)$$ Then $$z_1{\partial}_{z_1}(\phi_{\mu_1,\mu_2})=\left(\mu_1+\mu_2\frac{z_1}{z_1-z_2}\right)
\phi_{\mu_1,\mu_2},\eqno(2.22)$$ $$z_2{\partial}_{z_2}(\phi_{\mu_1,\mu_2})=\left(\mu_1-\mu_2\frac{z_2}{z_1-z_2}\right)
\phi_{\mu_1,\mu_2},\eqno(2.23)$$ $$(z_1{\partial}_{z_1})^2(\phi_{\mu_1,\mu_2})=\left(\mu_1^2+(2\mu_1+1)\mu_2\frac{z_1}
{z_1-z_2}+\mu_2(\mu_2-1)\frac{z_1^2}{(z_1-z_2)^2}\right)\phi_{\mu_1,\mu_2},\eqno(2.24)$$ $$(z_2{\partial}_{z_2})^2(\phi_{\mu_1,\mu_2})=\left(\mu_1^2-(2\mu_1+1)\mu_2\frac{z_2}{z_1-z_2}
+\mu_2(\mu_2-1)\frac{z_2^2}{(z_1-z_2)^2}\right)\phi_{\mu_1,\mu_2}.\eqno(2.25)$$ Thus $$[(z_1{\partial}_{z_1})^2+(z_2{\partial}_{z_2})^2](\phi_{\mu_1,\mu_2})-2\mu_2(\mu_2-1)\frac{z_1z_2}
{(z_1-z_2)^2}\phi_{\mu_1,\mu_2}=(2\mu_1^2+2\mu_1\mu_2+\mu_2^2)
\phi_{\mu_1,\mu_2},\eqno(2.26)$$ which is of the form (1.12) with $n=2$ and $$K=-2\mu_2(\mu_2-1),\qquad \nu=2\mu_1^2+2\mu_1\mu_2+\mu_2^2.\eqno(2.27)$$

Take $$g(z)=z^r\sum_{m=0}^{\infty}a_mz^m\qquad\mbox{with}\;\;r,a_m\in{\mathbb}{C}.\eqno(2.28)$$ Then $$z_1{\partial}_{z_1}\left(g\left(\frac{z_2}{z_2-z_1}\right)\right)=\frac{z_1z_2}{(z_2-z_1)^2}
g'\left(\frac{z_2}{z_2-z_1}\right),\eqno(2.29)$$ $$z_2{\partial}_{z_2}\left(g\left(\frac{z_2}{z_2-z_1}\right)\right)=-\frac{z_1z_2}{(z_2-z_1)^2}
g'\left(\frac{z_2}{z_2-z_1}\right),\eqno(2.30)$$ $$(z_1{\partial}_{z_1})^2\left(g\left(\frac{z_2}{z_2-z_1}\right)\right)=\frac{z_1^2z_2^2}
{(z_1-z_2)^4}g''\left(\frac{z_2}{z_2-z_1}\right)+\frac{z_1z_2(z_1+z_2)}{(z_2-z_1)^3}
g'\left(\frac{z_2}{z_2-z_1}\right),\eqno(2.31)$$ $$(z_2{\partial}_{z_2})^2\left(g\left(\frac{z_2}{z_1-z_2}\right)\right)=\frac{z_1^2z_2^2}
{(z_1-z_2)^4}g''\left(\frac{z_2}{z_2-z_1}\right)-\frac{z_1z_2(z_1+z_2)}{(z_1-z_2)^3}
g'\left(\frac{z_2}{z_2-z_1}\right).\eqno(2.32)$$ Hence by (2.22), (2.23), (2.26) and (2.29)-(2.32), we have $$\begin{aligned}
& &[(z_1{\partial}_{z_1})^2+(z_2{\partial}_{z_2})^2]\left (\phi_{\mu_1,\mu_2}
g\left(\frac{z_2}{z_2-z_1}\right)\right)\\ &=&\phi_{\mu_1,\mu_2}[\left(
2\mu_2(\mu_2-1)\frac{z_1z_2}{(z_1-z_2)^2}+(2\mu_1^2+2\mu_1\mu_2+\mu_2^2)\right)
g\left(\frac{z_2}{z_2-z_1}\right)\\
& &+
2\left(\mu_1+\mu_2\frac{z_1}{z_1-z_2}\right)\frac{z_1z_2}{(z_1-z_2)^2}g'\left(\frac{z_2}
{z_2-z_1}\right)
\\ &
&-2\left(\mu_1-\mu_2\frac{z_2}{z_1-z_2}\right)\frac{z_1z_2}{(z_1-z_2)^2}g'\left(\frac{z_2}
{z_2-z_1}\right)\\
&&+2\frac{z_1^2z_2^2}{(z_1-z_2)^4}g''\left(\frac{z_2}{z_2-z_1}\right)+2\frac{z_1z_2(z_1+z_2)}
{(z_2-z_1)^3}g'\left(\frac{z_2}{z_1-z_2}\right)]\\
&=& \phi_{\mu_1,\mu_2}[\left(
2\mu_2(\mu_2-1)\frac{z_1z_2}{(z_1-z_2)^2}+(2\mu_1^2+2\mu_1\mu_2+\mu_2^2)\right)g
\left(\frac{z_2}{z_2-z_1}\right)\\ &
&+2(1-\mu_2)\frac{z_1z_2(z_1+z_2)}{(z_2-z_1)^3}
g'\left(\frac{z_2}{z_1-z_2}\right)+2\frac{z_1^2z_2^2}{(z_1-z_2)^4}g''\left(\frac{z_2}{z_2-z_1}
\right)].\hspace{2.1cm}(2.33)\end{aligned}$$

Recall the classical Gauss hypergeometric equation $$x(1-x)\frac{d^2y}{dx^2}+[c-(a+b+1)x]\frac{dy}{dx}-aby=0,\eqno(2.34)$$ whose fundamental solutions are $$_2F_1(a,b;c; x)\qquad\mbox{and}\qquad x^{1-c}\: _2F_1(a+1-c,b+1-c;2-c; x)\eqno(2.35)$$ (see (1.17) and e.g. cf. \[AAR\]). Observe that $$\frac{z_1+z_2}{z_2-z_1}=2\frac{z_2}{z_2-z_1}-1,\;\;\frac{z_2}{z_2-z_1}\left(1-\frac{z_2}
{z_2-z_1}\right)=-\frac{z_1z_2}{(z_1-z_2)^2},\eqno(2.36)$$ Therefore, a sufficient condition for $\phi_{\mu_1,\mu_2}g(z_2/(z_2-z_1))$ to be a solution of the equation (1.12) with $n=2$ for suitable $K$ and $\nu$ is $$(1-\mu_2)\frac{z_1+z_2}{z_2-z_1}g'\left(\frac{z_2}{z_1-z_2}\right)+\frac{z_1z_2}{(z_1-z_2)^2}
g''\left(\frac{z_2}{z_2-z_1}\right)=\ell g\left(\frac{z_2}{z_2-z_1}\right)\eqno(2.37)$$ for some $\ell\in{\mathbb}{C}$, equivalently, $$(1-\mu_2)(2z-1)g'(z)-z(1-z)g''(z)=\ell g(z),\eqno(2.38)$$ which is of the form (2.34) with $$a+b+1=2(1-\mu_2),\;\;c=1-\mu_2,\;\;ab=-\ell.\eqno(2.39)$$ Take $a\in{\mathbb}{C}$ and $$b=1-2\mu_2-a,\qquad \ell=a(a+2\mu_2-1).\eqno(2.40)$$

The second fundamental solution of (2.38) is $$z^{\mu_2}\: _2F_1(a+\mu_2,1-\mu_2-a;\mu_2+1; z)\eqno(2.41)$$ (cf. (2.35)). In fact, by (2.20), $$\Psi(z_1,z_2)=(-1)^{\mu_2}\phi_{\mu_1,\mu_2}\left(\frac{z_2}{z_2-1}\right)^{\mu_2}\:
_2F_1\left(a+\mu_2,1-\mu_2-a;\mu_2+1;\frac{z_2}{z_2-1}\right)\eqno(2.42)$$ (cf. (1.14)) with $$\mu=\mu_2+a-1,\;\;{\lambda}_1=\mu_1-\frac{1}{2},\;\;{\lambda}_2=\mu_1+\mu_2+\frac{1}{2}.\eqno(2.43)$$ So the Etingof trace gives only one family of solutions of the form $\phi_{\mu_1,\mu_2}g(z_2/(z_2-z_1))$ of the equation (1.12) with $n=2$ and $$K=2\mu_2(1-\mu_2)+2a(a+2\mu_2-1),\qquad \nu=2\mu_1^2+2\mu_1\mu_2+\mu_2^2.\eqno(2.44)$$

<span>**Theorem 2.2**</span>. <span>*For $g(z)$ in (2.28), the product function $\phi_{\mu_1,\mu_2}(z_1,z_2)g(z_2/(z_2-z_1))$ is a solution of the equation (1.12) with $n=2$ and suitable constants $K,\mu$ if $g(z)$ is a solution of the classical hypergeometric equation (2.38). In particular, the functions $$(z_1z_2)^{\mu_1}(z_1-z_2)^{\mu_2}\;_2F_1(a,1-2\mu_2-a;1-\mu_2;z)\eqno(2.45)$$ for $a,\mu_1,\mu_2\in{\mathbb}{C}$ are a new family of solutions of the equation (1.12) with $n=2$ and $K,\nu$ in (2.44).*</span>

Etingof Trace for $gl(n)$ with $n>2$
====================================

In this section, we find the Etingof trace functions for the general case of $gl(n)$ with $n>2$ under a certain condition and their master hypergeometric functions.

In the Lie algebra $gl(n)$ given in (1.4) and (1.5), we choose $$\{E_{i,j}\mid 1\leq i<j\leq n\}\;\;\mbox{as positive root vectors}.\eqno(3.1)$$ In particular, we have $$\{E_{i,i+1}\mid i=1,2,...,n-1\}\;\;\mbox{as  positive simple root vectors}.\eqno(3.2)$$ Accordingly, $$\{E_{i,j}\mid 1\leq j<i\leq n\}\;\;\mbox{are negative root vectors}\eqno(3.3)$$ and $$\{E_{i+1,i}\mid i=1,2,...,n-1\}\;\;\mbox{are negative simple root vectors}.\eqno(3.4)$$

Recall ${\Gamma}_A$ in (1.19). Let $U(gl(n))$ be the universal enveloping algebra of $gl(n)$. For $${\alpha}=\sum_{1\leq j<i\leq n}{\alpha}_{ij}{\epsilon}_{i,j}\in {\Gamma}_A,\eqno(3.5)$$ we denote $$E^{\alpha}=E_{2,1}^{{\alpha}_{2,1}}E_{3,1}^{{\alpha}_{3,1}}E_{3,2}^{{\alpha}_{3,2}}E_{4,1}^{{\alpha}_{4,1}}\cdots
E_{n,1}^{{\alpha}_{n,1}}\cdots E_{n,n-1}^{{\alpha}_{n,n-1}}\in U(gl(n)).\eqno(3.6)$$ Denote by ${\cal G}_-$ the Lie subalgebra spanned by (3.3). Then $$\{E^{\alpha}\mid {\alpha}\in{\Gamma}_A\}\;\;\mbox{forms a PBW basis of}\;\;U({\cal G}_-).\eqno(3.7)$$

Let ${\lambda}$ be a weight of $gl(n)$ such that $${\lambda}_1-{\lambda}_2=\cdots ={\lambda}_{n-2}-{\lambda}_{n-1}=\mu\;\;\mbox{and}\;\;{\lambda}_{n-1}-{\lambda}_n
\not\in{\mathbb}{N},
\eqno(3.8)$$ for some constant $\mu$. Set $${\sigma}_i={\lambda}_i-{\lambda}_{i+1}\qquad{\mbox{for}}\;\;i\in{\overline}{1,n-1}.\eqno(3.9)$$ The Verma $gl(n)$-module with the highest-weight vector $v_{\lambda}$ of weight ${\lambda}$ is given by $$M_{\lambda}=\mbox{Span}\{E^{\alpha}v_{\lambda}\mid{\alpha}\in {\Gamma}_A\},\eqno(3.10)$$ with the action determined by $$\begin{aligned}
E_{i,i+1}(E^{\alpha}v_{\lambda})&=&(\sum_{p=1}^{i-1}{\alpha}_{i+1,p}E^{{\alpha}+{\epsilon}_{i,p}
-{\epsilon}_{i+1,p}}-\sum_{p=i+2}^n{\alpha}_{p,i}E^{{\alpha}+{\epsilon}_{p,i+1}-{\epsilon}_{p,i}}\\
& &+{\alpha}_{i+1,i}
({\sigma}_i+1-\sum_{p=i+1}^n{\alpha}_{p,i}+\sum_{p=i+2}^n{\alpha}_{p,i+1})E^{{\alpha}-{\epsilon}_{i+1,i}})v_{\lambda},
\hspace{2.2cm}(3.11)\end{aligned}$$ $$E_{j,i}(E^{\alpha}v_{\lambda})=(E^{{\alpha}+{\epsilon}_{j,i}}+\sum_{p=1}^{i-1}{\alpha}_{i,p}E^{{\alpha}+{\epsilon}_{j,p}
-{\epsilon}_{i,p}})
v_{\lambda},\eqno(3.12)$$ $$E_{k,k}(E^{\alpha}v_{\lambda})=({\lambda}_k+\sum_{p=1}^{k-1}{\alpha}_{k,p}-\sum_{q=k+1}^n{\alpha}_{q,k})
E^{\alpha}v_{\lambda}\eqno(3.13)$$ for $1\leq i<j\leq n$ and $k=1,2,...,n$.

Let $${\tilde}{M}=\{\sum_{i_1,\cdots,i_n\in{\mathbb}{Z}}v_{i_1,\cdots,i_n}x_1^{i_1+\mu}x_2^{i_2+\mu}
\cdots
 x_n^{i_n+\mu}\mid v_{i_1,\cdots,i_n}\in M_{\lambda}\},\eqno(3.14)$$ a space of Laurent series with coefficients in $M_{\lambda}$. Define the action of $gl(n)$ on ${\tilde}{M}$ by $$\begin{aligned}
& &E_{j_1,j_2}(\sum_{i_1,\cdots,i_n\in{\mathbb}{Z}}v_{i_1,\cdots,i_n}
x_1^{i_1+\mu}
x_2^{i_2+\mu}\cdots
x_n^{i_n+\mu})=\sum_{i_1,\cdots,i_n\in{\mathbb}{Z}}[E_{j_1,j_2}
(v_{i_1,\cdots,i_n})x_1^{i_1+\mu}x_2^{i_2+\mu}\cdots
x_n^{i_n+\mu}\\ & &+v_{i_1,\cdots,i_n}
(x_{j_1}{\partial}_{x_{j_2}}-\mu{\delta}_{j_1,j_2})(x_1^{i_1+\mu}x_2^{i_2+\mu}\cdots
x_n^{i_n+\mu})] \hspace{5.9cm}(3.15)\end{aligned}$$ for $j_1,j_2\in{\overline}{1,n}$. Then ${\tilde}{M}$ becomes a $gl(n)$-module that is isomorphic to $M_{\lambda}\hat{\otimes}V_\mu$, the completion of the tensor $M_{\lambda}\otimes V_\mu$ with respect to the topology given by weight subspaces (cf. (1.6) and (1.7)).

For $$\vec i=(i_1,i_2,...,i_{n-1})\in{\mathbb}{N}^{n-1},\eqno(3.16)$$ we denote $$x^{(\vec i)}=x_1^{i_1}x_2^{i_2-i_1}\cdots x_{n-1}^{i_{n-1}-i_{n-2}}x_n^{-i_{n-1}},
\qquad {\epsilon}(\vec i)=\sum_{p=1}^{n-1}i_p{\epsilon}_{p+1,p}.\eqno(3.17)$$ Then we have:

<span>**Lemma 3.1**</span>. <span>*The vector $$u_{\lambda}=(x_1x_2\cdots x_n)^{\mu}\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots
+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}E^{{\epsilon}(\vec i)} v_{\lambda}x^{(\vec i)}\eqno(3.18)$$ is a singular vector of weight ${\lambda}$, where ${\sigma}={\sigma}_{n-1}={\lambda}_{n-1}-{\lambda}_n$.*</span>

<span>*Proof*</span>. For $p\in\{1,2,...,n-2\}$, we have $$\begin{aligned}
 & &E_{p,p+1}(u_{\lambda})\\ &=&
(x_1x_2\cdots
x_n)^{\mu}\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}
{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}[(\mu+i_{p+1}-i_p) E^{{\epsilon}(\vec i)}v_{\lambda}x_px_{p+1}^{-1}x^{(\vec i)}\\ & &+(\mu+1-i_p+i_{p+1})i_p
E^{{\epsilon}(\vec i)-{\epsilon}_p}v_{\lambda}x^{(\vec i)}]\\ &=& (x_1x_2\cdots
x_n)^{\mu}[\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}
{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}(\mu+i_{p+1}-i_p) E^{{\epsilon}(\vec i)}v_{\lambda}x_px_{p+1}^{-1}x^{(\vec i)}\\ &&+
\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}} {i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}(\mu+1-i_p+i_{p+1})i_pE^{{\epsilon}(\vec i)-{\epsilon}_p}
v_{\lambda}x^{(\vec i)}]\hspace{3cm}\end{aligned}$$ $$\begin{aligned}
&=&(x_1x_2\cdots x_n)^{\mu}[\sum_{i_1,...,i_{n-1}=0}^{\infty}
\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}! {\langle}{\sigma}{\rangle}_{i_{n-1}}}(\mu+i_{p+1}-i_p)E^{{\epsilon}(\vec i)}v_{\lambda}x_px_{p+1}^{-1}x^{(\vec i)}
\\ &&-\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}
{\langle}\mu{\rangle}_{i_{n-1}}}
{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}(\mu+i_{p+1}-i_p)E^{{\epsilon}(\vec i)}v_{\lambda}x_p
x_{p+1}^{-1}x^{(\vec i)}]=0.\hspace{1.1cm}(3.19)\end{aligned}$$ Moreover, $$\begin{aligned}
& & E_{n-1,n}(u_{\lambda})\\ &=&
(x_1x_2\cdots
x_n)^{\mu}\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}[(\mu-i_{n-1})E^{{\epsilon}(\vec i)}v_{\lambda}x_{n-1}x_n^{-1}x^{(\vec i)}\\ &
&+({\sigma}+1-i_{n-1})i_{n-1}E^{{\epsilon}(\vec i)-{\epsilon}_{n-1}}v_{\lambda}x^{(\vec
i)}]\\ &=& (x_1x_2\cdots
x_n)^{\mu}[\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}(\mu-i_{n-1})E^{{\epsilon}(\vec i)}v_{\lambda}x_{n-1}x_n^{-1}x^{(\vec i)}\\ &&+
\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}({\sigma}+1-i_{n-1})i_{n-1}E^{{\epsilon}(\vec
i)-{\epsilon}_{n-1}}v_{\lambda}x^{(\vec i)}]\\ &=&(x_1x_2\cdots
x_n)^{\mu}[\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}(\mu-i_{n-1})E^{{\epsilon}(\vec i)}v_{\lambda}x_{n-1}x_n^{-1}x^{(\vec i)}\\
&&-\sum_{i_1,...,i_{n-1}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}{\langle}\mu{\rangle}_{i_{n-1}+1}}{i_1!i_2!\cdots i_{n-1}!{\langle}{\sigma}{\rangle}_{i_{n-1}}}E^{{\epsilon}(\vec i)}v_{\lambda}x_px_{p+1}^{-1}x^{(\vec
i)}]=0.\hspace{3.1cm}(3.20)\end{aligned}$$ Hence $u_{\lambda}$ is a singular vector. $\qquad\Box$

Let $\Phi: M_{\lambda}{\rightarrow}{\tilde}{M}$ be the Lie algebra module homomorphism such that $\Phi(v_{\lambda})=u_{\lambda}$. Let $\vec i$ as in (3.16) and ${\beta}\in{\Gamma}_A$. By (3.11), we have $$\begin{aligned}
& &E_{n,n-1}^{{\beta}_{n,n-1}}(E^{{\epsilon}(\vec i)}v_{\lambda})=
\sum_{p_{n,n-1}=0}^{{\beta}_{n,n-1}}
\left(\!\!\begin{array}{c}{\beta}_{n,n-1}\\ p_{n,n-1}\end{array}\!\!\right){\langle}i_{n-2}{\rangle}_{p_{n,n-1}}
\\ & &E^{{\epsilon}(\vec i)-p_{n,n-1}{\epsilon}_{n-1,n-2}+p_{n,n-1}{\epsilon}_{n,n-2}+({\beta}_{n,n-1}-p_{n,n-1})
{\epsilon}_{n,n-1}}v_{\lambda},\hspace{5cm}(3.21)\end{aligned}$$ $$\begin{aligned}
& &E_{n,n-2}^{{\beta}_{n,n-2}}E_{n,n-1}^{{\beta}_{n,n-1}}(E^{{\epsilon}(\vec i)}v_{\lambda})
=\sum_{p_{n,n-2},p_{n,n-1}=0}^{\infty}
\left(\!\!\begin{array}{c}{\beta}_{n,n-2}\\
p_{n,n-2}\end{array}\!\!\right)\left(\!\!\begin{array}
{c}{\beta}_{n,n-1}\\ p_{n,n-1}\end{array}\!\!\right){\langle}i_{n-3}{\rangle}_{p_{n,n-2}}{\langle}i_{n-2} {\rangle}_{p_{n,n-1}}\\ & &
E^{{\epsilon}(\vec
i)+\sum_{r=n-2,n-1}[-p_{n,r}{\epsilon}_{r,r-1}+p_{n,r}{\epsilon}_{n,r-1}+({\beta}_{n,r}-p_{n,r})
{\epsilon}_{n,r}]}v_{\lambda},\hspace{5.2cm}(3.22)\end{aligned}$$ $$\begin{aligned}
\qquad& &\left[\prod_{r=1}^{n-1}E_{n,r}^{{\beta}_{n,r}}\right]
(E^{{\epsilon}(\vec
i)}v_{\lambda})=\sum_{p_{n,2},...,,p_{n,n-1}=0}^{\infty}\left[\prod_{r=2}^{n-1}
\left(\!\!\begin{array}{c}{\beta}_{n,r}\\
p_{n,r}\end{array}\!\!\right){\langle}i_{r-1}{\rangle}_{p_{n,r}}\right]\\ & &
E^{{\epsilon}(\vec
i)+{\beta}_{n,1}{\epsilon}_{n,1}+\sum_{s=2}^{n-1}(-p_{n,s}{\epsilon}_{s,s-1}+p_{n,s}{\epsilon}_{n,s-1}
+({\beta}_{n,s}-p_{n,s}){\epsilon}_{n,s})}v_{\lambda},\hspace{3.8cm}(3.23)\end{aligned}$$ $$\begin{aligned}
&&\left[\prod_{r=1}^{n-2}E_{n-1,r}^{{\beta}_{n-1,r}}\right]\left[\prod_{r=1}^{n-1}
E_{n,r}^{{\beta}_{n,r}}\right](E^{{\epsilon}(\vec
i)}v_{\lambda})=\sum_{i=n-1,n}\;\sum_{j=1}^{i-1}
\sum_{p_{i,j}=0}^{\infty}\hspace{7cm}\end{aligned}$$ $$\begin{aligned}
 &
&\left[\prod_{r=1}^{n-2}\left(\!\!\begin{array}{c}{\beta}_{n-1,r}
\\ p_{n-1,r}\end{array}\!\!\right)\right]
\left[\prod_{r=1}^{n-1}\left(\!\!\begin{array}{c}{\beta}_{n,r}\\ p_{n,r}\end{array}\!\!\right)
\right]
\left[\prod_{r=1}^{n-3}{\langle}i_r{\rangle}_{p_{n,r+1}+p_{n-1,r+1}}\right]{\langle}i_{n-2}{\rangle}_{p_{n,n-1}}
\\ & &E^{{\epsilon}(\vec i)+\sum_{r=n-1,n}{\beta}_{r,1}{\epsilon}_{r,1}+\sum_{r=n-1,n}\sum_{s=2}^{r-1}(-p_{r,s}
{\epsilon}_{s,s-1}+p_{r,s}{\epsilon}_{r,s-1}+({\beta}_{r,s}-p_{r,s}){\epsilon}_{r,s})}v_{\lambda},\hspace{2.3cm}(3.24)\end{aligned}$$ $$\begin{aligned}
& &E^{\beta}(E^{{\epsilon}(\vec i)}v_{\lambda})=\sum_{2\leq k<j\leq n}\;
\sum_{p_{j,k}=0}^{\infty}
\left[\prod_{2\leq m<l\leq n}\left(\!\!\begin{array}{c}{\beta}_{l,m}\\
p_{l,m}\end{array}\!\!\right) \right]\left[\prod_{s=1}^{n-2}{\langle}i_s{\rangle}_{_{\sum_{r=s+2}^np_{r,s+1}}}\right]\\ & & E^{{\epsilon}(\vec
i)+\sum_{1\leq m<l\leq
n}({\beta}_{l,m}+(1-{\delta}_{l,m+1})p_{l,m+1}-p_{l,m}-{\delta}_{l,m+1}
\sum_{s=l+1}^np_{s,l}){\epsilon}_{l,m}}v_{\lambda}.\hspace{3.8cm}(3.25)\end{aligned}$$ where we treat $p_{2,1}=p_{3,1}=\cdots=p_{n,1}=0$. Hence by (3.14) and (3.25), we obtain $$\begin{aligned}
& &E^{\alpha}(E^{{\epsilon}(\vec i)}v_{\lambda}(x_1x_2\cdots x_n)^{\mu}x^{(\vec i)})\\
&=&\sum_{p_{2,1}^0,...,p_{n,1}^0=0}^{\infty} \;\sum_{2\leq k<j\leq
n}\;\sum_{p_{j,k}^0,p_{j,k}=0}^{\infty}\left[\prod_{m=2}^n\left(\!\!\begin{array}{c}{\alpha}_{m,1}\\
p_{m,1}^0\end{array}\!\!\right)\right]\left[\prod_{2\leq m<l\leq
n}\left(\!\!\begin{array}{c}{\alpha}_{l,m}\\
p_{l,m}^0,p_{l,m}\end{array}\!\!\right)\right]\\
&&\left[\prod_{s=1}^{n-2}{\langle}i_s{\rangle}_{_{\sum_{r=s+2}^np_{r,s+1}}}\right]\left[\prod_{s=1}^{n-1}{\langle}\mu+i_s-i_{s-1}{\rangle}_{_{\sum_{r=s+1}^np_{r,s}^0}}\right]\\ & &
E^{{\epsilon}(\vec i)+\sum_{1\leq m<l\leq
n}({\alpha}_{l,m}+p_{l,m+1}-p_{l,m}^0-p_{l,m}
-{\delta}_{l,m+1}\sum_{s=l+1}^np_{s,l}){\epsilon}_{l,m}}v_{\lambda}\\ & &
(x_1x_2\cdots
x_n)^{\mu}\prod_{r=1}^nx_r^{i_r-i_{r-1}+\sum_{s=1}^{r-1}p^0_{r,s}
-\sum_{q=r+1}^np_{q,r}^0}\hspace{6cm}(3.26)\end{aligned}$$ for ${\alpha}\in{\Gamma}_A$, where we treat $$i_{-1}=i_n=p_{2,1}=p_{3,1}=\cdots =p_{n,1}=p_{2,2}=p_{3,3}=\cdots p_{n,n}=0.\eqno(3.27)$$

View $\Phi$ as a function in $\{x_1,x_2,...,x_n\}$ taking values in $\mbox{End}\:M_{\lambda}$, the space of linear transformations on $M_{\lambda}$. The <span>*Etingof trace function*</span> $$E(z_1,z_2,...,z_n)=(x_1x_2\cdots x_n)^{-\mu}\mbox{tr}_{_{M_{\lambda}}}\:\Phi z_1^{E_{1,1}}
z_2^{E_{2,2}}\cdots z_n^{E_{n,n}}.\eqno(3.28)$$ In order to calculate $E(z_1,z_2,...,z_n)$, we need to take $$p_{l,m+1}=p_{l,m}^0+p_{l,m},\;\; i_k=p_{k+1,k}^0+p_{k+1,k}+\sum_{s=k+2}^np_{s,k+1}\eqno(3.29)$$ in (3.26) for $1\leq m<l-1\leq n-1$ and $k\in{\overline}{1,n-1}$. By (3.27), (3.29) and induction, $$p_{l,m}=\sum_{r=1}^{m-1}p_{l,r}^0\qquad{\mbox{for}}\;\;1\leq m<l\leq n.\eqno(3.30)$$ Moreover, $$i_k=\sum_{r=k+1}^n(p_{r,k}^0+p_{r,k})=\sum_{r=k+1}^n\sum_{s=1}^kp^0_{r,s}\qquad{\mbox{for}}\;\;
k\in{\overline}{1,n-1}.\eqno(3.31)$$ $$i_k-i_{k-1}=\sum_{r=k+1}^n\sum_{s=1}^kp^0_{r,s}-\sum_{r=k}^n\sum_{s=1}^{k-1}p^0_{r,s}=
\sum_{r=k+1}^np^0_{r,k}-\sum_{s=1}^{k-1}p^0_{k,s}\eqno(3.32)$$ for $k\in{\overline}{1,k}$. Furthermore, $$\left(\!\!\begin{array}{c}{\alpha}_{j,k}\\ p_{j,k}^0,p_{j,k}\end{array}\!\!\right)=\left(\!\!
\begin{array}{c}p^0_{j,k}+p_{j,k}\\ p_{j,k}\end{array}\!\!\right)\left(\!\!\begin{array}{c}
{\alpha}_{j,k}\\ p^0_{j,k}+p_{j,k}\end{array}\!\!\right) \eqno(3.33)$$ and $$\left(\!\!\begin{array}{c}p^0_{j,k}+p_{j,k}\\ p_{j,k}\end{array}\!\!\right)=
\frac{(p^0_{j,k}+p_{j,k})!}{p^0_{j,k}!p_{j,k}!}=\frac{(p^0_{j,k}+p_{j,k})!}{p^0_{j,k}!
(p^0_{j,k-1}+p_{j,k-1})!}\eqno(3.34)$$ by (3.29).

For convenience, we denote $${\alpha}_{_{{\underline}{1}}}={\alpha}_{{\overline}{n}}=0,\;\;{\alpha}_{_{{\underline}{i}}}=\sum_{r=1}^{i-1}{\alpha}_{i,r},\;\;
{\alpha}_{_{{\overline}{j}}}=\sum_{s=j+1}^n{\alpha}_{s,j}\eqno(3.35)$$ for ${\alpha}\in{\Gamma}_A,\;i\in {\overline}{2,n}$ and $j\in{\overline}{1,n-1}$. By (2.16), (3.18) and (3.26)-(3.35), the Etingof's trace function $$\begin{aligned}
& &E(z_1,z_2,...,z_n)
\\ &=&\left(\prod_{r=1}^nz_r^{{\lambda}_r}\right)\sum_{{\alpha}\in{\Gamma}_A}\;\sum_{p_{r_1,r_2}^0\in{\mathbb}{N},
\;1\leq r_2<r_1\leq n}(-1)^{\sum_{s=1}^{n-1}i_s}\left[\prod_{2\leq
k<j\leq n}\left(\!\!
\begin{array}{c}p^0_{j,k}+p_{j,k}\\ p_{j,k}\end{array}\!\!\right)\right]\\ &&\times
\frac{\prod_{s=1}^{n-1}{\langle}\mu+i_s-i_{s-1}{\rangle}_{_{\sum_{r=s+1}^np^0_{r,s}}}}
{{\langle}{\sigma}{\rangle}_{_{i_{n-1}}}\prod_{s=1}^{n-1}(p_{s+1,s}^0+p_{s+1.s})!}
\left[\prod_{1\leq k<j\leq n}\left(\!\!\begin{array}{c}{\alpha}_{j,k}\\ p^0_{j,k}+p_{j,k}
\end{array}\!\!\right)\right] {\langle}\mu{\rangle}_{i_{n-1}}
\prod_{1\leq k<j\leq n}\left(\frac{z_j}{z_k}\right)^{{\alpha}_{j,k}}\\
&=&\left(\prod_{r=1}^nz_r^{{\lambda}_r}\right)\sum_{p_{r_1,r_2}^0\in{\mathbb}{N},\;1\leq
r_2<r_1\leq
n}(-1)^{\sum_{s=1}^{n-1}i_s}\frac{\prod_{s=1}^{n-1}{\langle}\mu+i_s-i_{s-1}{\rangle}_{_{\sum_{r=s+1}^n
p^0_{r,s}}}{\langle}\mu{\rangle}_{i_{n-1}}}{{\langle}{\sigma}{\rangle}_{_{i_{n-1}}}\prod_{1\leq
s_1<s_2\leq n}p_{s_2,s_1}^0!}\\ & &\times \left[\prod_{1\leq
r_1<r_2\leq n}\left(\frac{z_{r_2}}{z_{r_1}}\right)^{p_{r_2,r_1}^0
+p_{r_2,r_1}}\right]\left[\prod_{1\leq s_1<s_2\leq
n}\left(\frac{1}{1-z_{s_2}/z_{s_1}}
\right)^{p_{s_2,s_1}^0+p_{s_2,s_1}+1}\right] \\
&=&\frac{\prod_{r=1}^nz_r^{{\lambda}_r+n-r}}{\prod_{1\leq k<j\leq
n}(z_k-z_j)}\sum_{{\beta}\in{\Gamma}_A}\frac{\left[\prod_{s=1}^{n-1}{\langle}\mu+{\beta}_{{\overline}{s}}-{\beta}_{{\underline}{s}}
{\rangle}_{_{{\beta}_{{\overline}{s}}}}\right]{\langle}\mu{\rangle}_{_{{\beta}_{{\underline}{n}}}}}{{\langle}{\sigma}{\rangle}_{_{{\beta}_{{\underline}{n}}}}
\prod_{1\leq s_1<s_2\leq n}{\beta}_{s_2,s_1}!}\\ & &\times
\left[\prod_{1\leq k<j\leq
n}\left(\frac{-z_k}{z_k-z_j}\right)^{\sum_{r=1}^k{\beta}_{j,r}}\right].\hspace{8.2cm}(3.36)\end{aligned}$$ Set $$\xi_{r_2,r_1}^A=\prod_{s=r_1}^{r_2-1}\frac{z_{r_2}}{z_{r_2}-z_s}
\qquad {\mbox{for}}\;\;1\leq r_1<r_2\leq n.\eqno(3.37)$$

Motivated by (2.17)-(2.19) and (3.36), we define our $(n(n-1)/2)$-variable hypergeometric function of type A by $${\cal X}_A(\tau_1,..,\tau_n;{\vartheta})\{z_{j,k}\}=\sum_{{\beta}\in{\Gamma}_A}\frac{\left[\prod_{s=1}^{n-1}
(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right](\tau_n)_{_{{\beta}_{{\underline}{n}}}}}
{{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}z^{\beta}\eqno(3.38)$$ (cf. (3.35)), where $${\beta}!=\prod_{1\leq k<j\leq n}{\beta}_{j,k}!,\qquad z^{\beta}=\prod_{1\leq k<j\leq n}z_{j,k}^{{\beta}_{j,k}}.
\eqno(3.39)$$ According to (2.18) and (3.36)-(3.38), we have:

<span>**Theorem 3.3**</span>. <span>*Under the condition (3.8), the Etingof trace function is*</span> $$E(z_1,z_2,...,z_n)=\left[\frac{\prod_{r=1}^nz_r^{{\lambda}_r+n-r}}{\prod_{1\leq k<j\leq n}
(z_k-z_j)}\right]{\cal
X}_A(\mu+1,..,\mu+1,-\mu;-{\sigma})\{\xi_{r_2,r_1}^A\}.\eqno(3.40)$$

The function in (1.14) $$\Psi(z_1,z_2,\cdots,z_n)=\prod_{r=1}^nz_r^{{\lambda}_r+(n+1)/2-r}
{\cal
X}_A(\mu+1,..,\mu+1,-\mu;-{\sigma})\{\xi_{r_2,r_1}^A\}.\eqno(3.41)$$

Etingof Trace for $sp(2n)$
==========================

In this section, we want to introduce and calculate the Etingof trace function of $sp(2n)$.

The symplectic Lie algebra $$\begin{aligned}
\hspace{1cm}sp(2n)&=&\sum_{i,j=1}^n{\mathbb}{C}(E_{i,j}-E_{n+j,n+i})
+\sum_{i=1}^n({\mathbb}{C}E_{i,n+i}+{\mathbb}{C}E_{n+i,i})\\  & &+\sum_{1\leq i<j\leq n }[{\mathbb}{C}
(E_{i,n+j}+E_{n+j,i})+{\mathbb}{C}(E_{n+i,j}+E_{n+j,i})]\hspace{3cm}(4.1)\end{aligned}$$ is a Lie subalgebra of $gl(2n)$. Set $$h_n=E_{n,n}-E_{2n,2n},\;\;h_i=E_{i,i}-E_{i+1,i+1}-E_{n+i,n+i}+E_{n+i+1,n+i+1}\eqno(4.2)$$ for $i=1,2,...,n-1.$ The subspace $$T=\sum_{i=1}^n{\mathbb}{C}h_i\eqno(4.3)$$ forms a Cartan subalgebra of $sp(2n)$. We choose positive root vectors $$\{E_{i,j}-E_{n+j,n+i},\;E_{i,n+j}+E_{j,n+i},\;E_{k,n+k} \mid 1\leq i<j\leq n,\; k{\overline}{1,n}\}.
\eqno(4.4)$$ In particular, we take $$\{E_{i,i+1}-E_{n+i+1,n+i},\;E_{n,2n}\mid i=1,2,...,n-1\}\;\;\mbox{as positive simple root
vectors}.\eqno(4.5)$$ Accordingly, $$\{E_{i,j}-E_{n+j,n+i},\;E_{i,n+j}+E_{j,n+i},\;E_{n+k,k} \mid 1\leq j<i\leq n,\;k\in{\overline}{1,n}\}
\eqno(4.6)$$ are negative root vectors and $$\{E_{i+1,i}-E_{n+i,n+i+1},\;E_{2n,n}\mid i\in{\overline}{1,n-1}\}\;\;\mbox{are negative simple root
vectors}.\eqno(4.7)$$

For convenience, we denote $$C_{i,j}=E_{i,j}-E_{n+j,n+i}\qquad{\mbox{for}}\;\;i,j\in{\overline}{1.n},\eqno(4.8)$$ $$C_{j,n+k}=E_{j,n+k}+E_{k,n+j},\qquad C_{n+j,k}=E_{n+j,k}+E_{n+k,j}\eqno(4.9)$$ for $j,k\in{\overline}{1,n}$ such that $j\neq k$, and $$C_{r,n+r}=E_{r,n+r},\qquad C_{n+r,r}=E_{n+r,r}\qquad{\mbox{for}}\;\;r\in{\overline}{1,n}.\eqno(4.10)$$ Let $${\Gamma}_C=\sum_{1\leq j<i\leq n}{\mathbb}{N}{\epsilon}_{i,j}+\sum_{1\leq j\leq i\leq n}{\mathbb}{N}{\epsilon}_{n+i,j}
\eqno(4.11)$$ be the torsion-free additive semigroup of rank $n^2$ with ${\epsilon}_{p,q}$ as base elements, and let $U(sp(2n))$ be the universal enveloping algebra of $sp(2n)$. For any $${\alpha}=\sum_{1\leq j<i\leq n}{\alpha}_{i,j}{\epsilon}_{i,j}+\sum_{1\leq j\leq i\leq n}{\alpha}_{n+i,j}{\epsilon}_{n+i,j}
\in {\Gamma}_C,\eqno(4.12)$$ we denote $$\begin{aligned}
\hspace{1cm}C^{\alpha}&=&C_{2,1}^{{\alpha}_{2,1}}C_{3,1}^{{\alpha}_{3,1}}C_{3,2}^{{\alpha}_{3,2}}
C_{4,1}^{{\alpha}_{4,1}}\cdots C_{n,1}^{{\alpha}_{n,1}}\cdots C_{n,n-1}^{{\alpha}_{n,n-1}}\\ & &\times
 C_{n+1,1}^{{\alpha}_{n+1,1}}C_{n+2,1}^{{\alpha}_{n+2,1}}C_{n+2,2}^{{\alpha}_{n+2,2}}C_{n+3,1}^{{\alpha}_{n+3,1}}
 \cdots C_{2n,1}^{{\alpha}_{2n,1}}\cdots C_{2n,n}^{{\alpha}_{2n,n}}\hspace{3.6cm}(4.13)\end{aligned}$$ Denote by ${\cal G}_-$ the Lie subalgebra spanned by (4.6). Then $$\{C^{\alpha}\mid {\alpha}\in{\Gamma}_C\}\;\;\mbox{forms a PBW basis of}\;\;U({\cal G}_-)).\eqno(4.14)$$

For convenience, we treat $${\epsilon}_{n+i,j}={\epsilon}_{n+j,i}\qquad{\mbox{for}}\;\;i,j=1,2,...,n.\eqno(4.15)$$ Note that $$[C_{n,2n},C_{2n,n}]=h_n,\qquad [C_{i,i+1},C_{i+1,i}]=h_i,\eqno(4.16)$$ $$[h_i,C_{n+i+1,i}]=0,\qquad [h_i,C_{n+i,i}]=-2C_{n+i,i},\qquad [h_i,C_{n+i+1,i+1}]=2C_{n+i+1,i+1},\eqno(4.17)$$ $$[h_i,C_{n+k,i}]=-C_{n+k,i},\qquad[h_i,C_{n+k,i+1}]=C_{n+k,i+1},\eqno(4.18)$$ $$[C_{i,i+1},C_{n+p,i}]=-C_{n+p,i+1},\qquad [C_{i,i+1},C_{n+i+1,i}]=-2C_{n+i+1,i+1},\eqno(4.19)$$ $$[C_{n,2n},C_{2n,i}]=C_{n,i},\qquad [C_{n+j,r}, C_{r,q}]=(1+{\delta}_{j,q})C_{n+j,q},
\qquad[C_{n+r,r},C_{r,q}]=C_{n+r,q}\eqno(4.20)$$ for $i\in{\overline}{1,n-1}$ and $ j,k,p,q,r\in{\overline}{1,n}$ such that $k\neq i,i+1,\;p\neq i+1$ and $q< r$.

Let ${\lambda}$ be a weight, which is a linear function on $T$, such that $${\lambda}(h_i)=-\frac{1}{2}\qquad{\mbox{for}}\;\;i=1,2,...,n-1;\qquad {\lambda}(h_n)={\lambda}_n\in{\mathbb}{C}. \eqno(4.21)$$ Recall that $sp(2n)$ is generated by $$\{C_{i,i+1},C_{i+1,i},C_{n,2n},C_{2n,n},\mid i=1,2,...,n-1\}\eqno(4.22)$$ as a Lie algebra. The Verma $sp(2n)$-module with the highest-weight vector $v_{\lambda}$ of weight ${\lambda}$ is given by $$M_{\lambda}=\mbox{Span}\{E^{\alpha}v_{\lambda}\mid{\alpha}\in {\Gamma}_C\},\eqno(4.23)$$ with the action determined by $$\begin{aligned}
& &C_{i,i+1}(C^{\alpha}v_{\lambda})\\ &=&[\sum_{j=1}^{i-1}{\alpha}_{i+1,j}C^{{\alpha}+{\epsilon}_{i,j}
-{\epsilon}_{i+1,j}}-\sum_{j=i+2}^n{\alpha}_{j,i}C^{{\alpha}+{\epsilon}_{j,i+1}-{\epsilon}_{j,i}}-\sum_{k\neq i+1}{\alpha}_{n+k,i}
C^{{\alpha}-{\epsilon}_{n+k,i}+{\epsilon}_{n+k,i+1}}\\ & &-2{\alpha}_{n+i+1,i}C^{{\alpha}-{\epsilon}_{n+i+1,i}+{\epsilon}_{n+i+1,i+1}}+
{\alpha}_{i+1,i}(1/2-\sum_{j=i+1}^n{\alpha}_{j,i}+\sum_{j=i+2}^n{\alpha}_{j,i+1}\\ & &+\sum_{k\neq i,i+1}
({\alpha}_{n+k,i+1}-{\alpha}_{n+k,i})-2{\alpha}_{n+i,i}+2{\alpha}_{n+i+1,i+1})C^{{\alpha}-{\epsilon}_{i+1,i}}]v_{\lambda}\hspace{2.6cm}(4.24)\end{aligned}$$ for $i\in {\overline}{1,n-1}$, $$\begin{aligned}
& &C_{n,2n}(C^{\alpha}v_{\lambda})=[\sum_{i=1}^{n-1}{\alpha}_{2n,i}(C^{{\alpha}-{\epsilon}_{2n,i}
+{\epsilon}_{n,i}}+({\alpha}_{2n,i}-1)C^{{\alpha}-2{\epsilon}_{2n,i}+{\epsilon}_{n+i,i}}\\ & &+\sum_{j=1}^{i-1}{\alpha}_{2n,j}
C^{{\alpha}-{\epsilon}_{2n,i}-{\epsilon}_{2n,j}+{\epsilon}_{n+i,j}})+{\alpha}_{2n,n}({\lambda}_n+1-{\alpha}_{2n,n})C^{{\alpha}-{\epsilon}_{2n,n}}]
v_{\lambda}\hspace{2.5cm}(4.25)\end{aligned}$$ and $$C_{j,i}(E^{\alpha}v_{\lambda})=(C^{{\alpha}+{\epsilon}_{j,i}}+\sum_{p=1}^{i-1}{\alpha}_{i,p}C^{{\alpha}+{\epsilon}_{j,p}
-{\epsilon}_{i,p}})v_{\lambda},\eqno(4.26)$$ $$\begin{aligned}
C_{n+j,i}(E^{\alpha}v_{\lambda})&=&(C^{{\alpha}+{\epsilon}_{n+j,i}}+\sum_{q=1}^{j-1}{\alpha}_{n+j,q}
C^{{\alpha}+{\epsilon}_{n+q,i}-{\epsilon}_{j,q}}\\ & &+
\sum_{p=1}^{i-1}{\alpha}_{i,p}(C^{{\alpha}+{\epsilon}_{n+j,p}-{\epsilon}_{i,p}}+\sum_{q=1}^{j-1}{\alpha}_{n+j,q}C^{{\alpha}+
{\epsilon}_{n+q,p}-{\epsilon}_{i,p}-{\epsilon}_{j,q}})\hspace{2cm}(4.27)\end{aligned}$$ for $1\leq i<j\leq n$, $$\begin{aligned}
\hspace{1cm}C_{n+k,k}(C^{\alpha}v_{\lambda})&=&[C^{{\alpha}+{\epsilon}_{n+k,k}}+\sum_{r=1}^{k-1}
{\alpha}_{k,r}(C^{{\alpha}-{\epsilon}_{k,r}+{\epsilon}_{n+k,r}}+({\alpha}_{k,r}-1)C^{{\alpha}-2{\epsilon}_{k,r}+{\epsilon}_{n+r,r}}\\ &&
+\sum_{s=r+1}^{k-1}{\alpha}_{k,s}C^{{\alpha}-{\epsilon}_{k,r}-{\epsilon}_{k,s}+{\epsilon}_{n+s,r}})]v_{\lambda}\hspace{4.6cm}
(4.28)\end{aligned}$$ by (4.16)-(4.20).

Let $\{x_1,x_2,...,x_n\}$ be $n$ indeterminates. Set $$x^{\vec i}=x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n}\qquad\vec i=(i_1,i_2,...,i_n)\in{\mathbb}{Z}^n
\eqno(4.29)$$ and $$x^\ast=x_1^{-1/2}x_2^{-1/2}\cdots x_n^{-1/2}.\eqno(4.30)$$ Denote by $${\tilde}{M}=\{\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast\mid w_{\vec i}\in M_{\lambda}\},
\eqno(4.31)$$ the space of formal Laurent series in $\{x_1,x_2,...,x_n\}$ with the coefficients in $M_{\lambda}$. Moreover, we define the action of $sp(2n)$ on ${\tilde}{M}$ by $$C_{p,q}(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=\sum_{\vec i\in{\mathbb}{Z}^n}
\left(C_{p,q}(w_{\vec i})x^{\vec i}x^\ast+w_{\vec i}x_p{\partial}_{x_q}(x^{\vec i}x^\ast)
+\frac{{\delta}_{p,q}}{2}w_{\vec i}x^{\vec i}x^\ast\right),\eqno(4.32)$$ $$C_{p,n+q}(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=\sum_{\vec i\in{\mathbb}{Z}^n}
\left(C_{p,n+q}(w_{\vec i})x^{\vec i}x^\ast-\frac{1}{1+{\delta}_{p,q}}w_{\vec i}x_px_qx^{\vec i}
x^\ast\right),\eqno(4.33)$$ $$C_{n+p,q}(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=\sum_{\vec i\in{\mathbb}{Z}^n}
(C_{n+p,q}(w_{\vec i})x^{\vec i}x^\ast+\frac{1}{1+{\delta}_{p,q}}w_{\vec i}{\partial}_{x_p}{\partial}_{x_q}
(x^{\vec i}x^\ast))\eqno(4.34)$$ for $1\leq p,q\leq n$. It can be verified that ${\tilde}{M}$ forms an $sp(2n)$-module (cf. \[F\]).

For $\vec i\in {\mathbb}{N}\:^n$, we set $$x^{(\vec i)}=x_1^{i_1}x_2^{i_2-i_1}x_3^{i_3-i_2}\cdots x_{n-1}^{i_{n-1}-i_{n-2}}x_n^{2i_n
-i_{n-1}},\eqno(4.35)$$ $${\epsilon}(\vec i)=\sum_{p=1}^{n-1}i_p{\epsilon}_{p+1,p}+i_n{\epsilon}_{2n,n}.\eqno(4.36)$$ Let $$u_{\lambda}=\sum_{\vec i\in{\mathbb}{N}\:^n}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}}{i_1!i_2!\cdots i_n!
2^{i_n}{\langle}{\lambda}_n{\rangle}_{i_n}}C^{{\epsilon}(\vec i)}v_{\lambda}x^{(\vec i)}x^\ast.\eqno(4.37)$$ It is verified that $u_{\lambda}$ is a singular vector of weight ${\lambda}$ in ${\tilde}{M}$, that is, $$C_{i,j}(u_{\lambda})=C_{p,n+q}(u_{\lambda})=0\qquad{\mbox{for}}\;\;1\leq i<j\leq n;\;1\leq p\leq q\leq n.
\eqno(4.38)$$ Hence there exists $sp(2n)$-module homomorphism $\Phi: M_{\lambda}{\rightarrow}{\tilde}{M}$ such that $$\Phi(v_{\lambda})=u_{\lambda}.\eqno(4.39)$$ View $\Phi$ as a function in $\{x_1,x_2,...,x_n\}$ taking value in the spaces of linear transformations on $M_{\lambda}$. The Etingof trace function is defined as $$E_C(z_1,z_2,...,z_n)=(x^\ast)^{-1}\mbox{tr}_{M_{\lambda}}\Phi z_1^{C_{1,1}} z_2^{C_{2,2}}\cdots
 z_n^{C_{n,n}}.\eqno(4.40)$$ In the rest of this section, we want to calculate $E_C(z_1,z_2,...,z_n)$.

For $\vec i\in{\mathbb}{N}\:^n$ and ${\beta}\in {\Gamma}_C$ (cf. (4.11)), we have $$\begin{aligned}
& &C^{\beta}(C^{{\epsilon}(\vec i)}v_{\lambda})=\sum_{2\leq j_1<j_2\leq n}\;\sum_{k_1=1}^n
\;\sum_{k_2=2}^n\;\sum_{k_3=k_2}^n(\sum_{p_{j_2,j_1},q_{k_1,k_2},q_{k_2,k_2}^1=0}^{\infty}\\
& &\left[\prod_{2\leq l_1<l_2\leq
n}\left(\!\!\begin{array}{c}{\beta}_{l_2,l_1}\\
p_{l_2,l_1}\end{array}\!\!\right)\left(\!\!\begin{array}{c}{\beta}_{n+l_2,l_1}\\
q_{l_1,l_2},q_{l_2,l_1},
q^1_{l_2,l_1}\end{array}\!\!\right)\right]
\left[\prod_{r=2}^n\left(\!\!\begin{array}{c}{\beta}_{n+r,r}\\
q_{r,r},q_{r,r}^1\end{array}\!\!\right)\left(\!\!\begin{array}{c}{\beta}_{m+r,1}\\
q_{1,r}\end{array}\!\!\right)\right]\\ & &\times
\left[\prod_{s=1}^{n-2}{\langle}i_s{\rangle}_{_{\sum_{r=s+2}^np_{r,s+1}+\sum_{r=1}^nq_{r,s+1}+\sum_{r=2}^{s+1}q^1_{s+1,r}+
\sum_{r=s+1}^nq^1_{r,s+1}
}}\right]{\langle}i_{n-1}{\rangle}_{_{\sum_{r=1}^nq_{r,n}+\sum_{r=2}^nq^1_{n,r}+q_{n,n}^1}}
\\ & &\times
E^{\sum_{1\leq l_1<l_2\leq
n}(p_{l_2,l_1+1}-p_{l_2,l_1}-{\delta}_{l_2,l_1+1}(\sum_{r=l_2+1}^n
p_{r,l_2}+\sum_{s=1}^nq_{s,l_2}+\sum_{s=2}^{l_2}q^1_{l_2,s}+\sum_{s=l_2}^nq^1_{s,l_2}))
{\epsilon}_{l_2,l_1}}
\\ & &\times E^{\sum_{1\leq s_1<s_2\leq n}(q_{s_2,s_1+1}+q_{s_1,s_2+1}+q_{s_2+1,s_1+1}^1
-q_{s_1.s_2}-q_{s_2,s_1}-q_{s_2,s_1}^1){\epsilon}_{n+s_2,s_1}}\\ &
&\times E^{{\epsilon}(\vec i)+{\beta}+\sum_{r=1}^n(q_{r,r+1}+q_{r+1,r+1}^1-q_{r,r}-q_{r,r}^1){\epsilon}_{n+r,r}}v_{\lambda})\hspace{6.4cm}
(4.41)\end{aligned}$$ by the calculations in (3.21)-(3.26) and (4.26)-(4.28), where we treat $$p_{r,r}=q_{1,1}=q^1_{n+1,n+1}=q^1_{r,1}=q^1_{n+1,r}=q_{s,n+1}=q_{n+1,s}=0\eqno(4.42)$$ for $2\leq r\leq n$ and $1\leq s\leq n.$ Hence by (4.32), (4.34) and (4.41), we obtain $$\begin{aligned}
& &C^{\alpha}(C^{{\epsilon}(\vec i)}v_{\lambda}x^{(\vec i)}x^\ast)\\ &=&
\sum_{2\leq j_1<j_2\leq
n}\;\sum_{k_1=1}^n\;\sum_{k_2=2}^n\;\sum_{k_3=k_1}^n\;\sum_{k_4=k_2}^n\{\sum_{p^0_{k_2,1},
p_{j_2,j_1}^0,p_{j_2,j_1},q^0_{k_3,k_1},q^1_{k_3,k_1},q_{k_1,k_2}=0}^{\infty}\frac{1}
{2^{\sum_{r=1}^nq^0_{r,r}}}\\
& &\times \left[\prod_{2\leq l_1<l_2\leq
n}\left(\!\!\begin{array}{c}{\alpha}_{l_2,l_1}\\  p_{l_2,l_1}^0,
p_{l_2,l_1}\end{array}\!\!\right)\left(\!\!\begin{array}{c}{\alpha}_{n+l_2,l_1}\\
q_{l_2,l_1}^0,q_{l_2,l_1}^1,
q_{l_1,l_2},q_{l_2,l_1}\end{array}\!\!\right)\right]\\ &
&\times\left[\prod_{m=2}^n\left(\!\!\begin{array}{c}{\alpha}_{m,1}\\
p_{m,1}^0\end{array}\!\!\right)\right]\left(\!\!\begin{array}{c}{\alpha}_{n+1,1}\\
q_{1,1}^0\end{array}\!\!\right)\left[\prod_{r=2}^n\left(\!\!\begin{array}{c}{\alpha}_{n+r,r}\\
q_{r,r}^0,
q_{r,r}^1,q_{r,r}\end{array}\!\!\right)\left(\!\!\begin{array}{c}{\alpha}_{n+r,1}\\
q_{r,1}^0,q_{1,r}\end{array}\!\!\right)\right]\\ & &\times
\left[\prod_{s=1}^{n-2}{\langle}i_s{\rangle}_{_{\sum_{r=s+2}^np_{r,s+1}+\sum_{r=1}^nq_{r,s+1}+\sum_{r=2}^{s+1}q^1_{s+1,r}+
\sum_{r=s+1}^nq^1_{r,s+1}
}}\right]{\langle}i_{n-1}{\rangle}_{_{\sum_{r=1}^nq_{r,n}+\sum_{r=2}^nq^1_{n,r}+q_{n,n}^1}}\\
& &\times \left[\prod_{s=1}^{n-1}{\langle}i_s-i_{s-1}-1/2{\rangle}_{_{\sum_{r=s+1}^np_{r,s}^0+\sum_{r=1}^sq_{s,r}^0+\sum_{r=s}^nq_{r,s}^0}}
\right]{\langle}2i_n-i_{n-1}-1/2{\rangle}_{_{2q^0_{n,n}+\sum_{r=1}^{n-1}q^0_{n,r}}}
\\ & &\times
E^{\sum_{1\leq l_1<l_2\leq
n}(p_{l_2,l_1+1}-p_{l_2,l_1}-p_{l_2,l_1}^0-{\delta}_{l_2,l_1+1}
(\sum_{r=l_2+1}^np_{r,l_2}+\sum_{s=1}^nq_{s,l_2}+\sum_{s=2}^{l_2}q^1_{l_2,s}+\sum_{s=l_2}^n
q^1_{s,l_2})){\epsilon}_{l_2,l_1}}
\\ & &\times E^{\sum_{1\leq s_1<s_2\leq n}(q_{s_2,s_1+1}+q_{s_1,s_2+1}+q_{s_2+1,s_1+1}^1
-q_{s_1.s_2}-q_{s_2,s_1}-q_{s_2,s_1}^0-q_{s_2,s_1}^1){\epsilon}_{n+s_2,s_1}}\\ & &\times
E^{{\alpha}+{\epsilon}(\vec i)+\sum_{r=1}^n(q_{r,r+1}+q_{r+1,r+1}^1-q_{r,r}-q_{r,r}^0-q_{r,r}^1)
{\epsilon}_{n+r,r}}v_{\lambda}\\ & &
\left[\prod_{s=1}^{n-1}x_s^{i_s-i_{s-1}-1/2+\sum_{r=1}^{s-1}p^0_{s,r}-\sum_{r=s+1}^np_{r,s}^0
-\sum_{r=1}^sq_{s,r}^0-\sum_{r=s}^nq_{r,s}^0}\right]\\ & &\times x_n^{2i_n-i_{n-1}-1/2+
\sum_{r=1}^{n-1}p^0_{n,r}-2q^0_{n,n}-\sum_{r=1}^{n-1}q^0_{n,r}}\},\hspace{6.3cm}(4.43)\end{aligned}$$ where we treat $i_{-1}=0$.

In order to calculate the Etingof trace (4.40), we have to let in (4.43): $$p_{r_2,r_1}=p_{r_2,r_1-1}^0+p_{r_2,r_1-1}\qquad{\mbox{for}}\;\;2\leq r_1<r_2\leq n,\eqno(4.44)$$ $$q_{r,r+1}+q_{r+1,r+1}^1=q^0_{r,r}+q_{r,r}++q_{r,r}^1\qquad{\mbox{for}}\;\;r\in{\overline}{1,n-1},\eqno(4.45)$$ $$q_{s_2,s_1+1}+q_{s_1,s_2+1}+q_{s_2+1,s_1+1}^1=q_{s_1.s_2}+q_{s_2,s_1}+q_{s_2,s_1}^0
+q_{s_2,s_1}^1\eqno(4.46)$$ for $1\leq s_1<s_2\leq n$ and $$i_r=p^0_{r+1,r}+p_{r+1,r}+\sum_{s=r+2}^np_{s,r+1}+\sum_{s=1}^nq_{s,r+1}
+\sum_{s=2}^{r+1}q_{r+1,s}^1+\sum_{s=r+1}^nq_{s,r+1}^1\eqno(4.47)$$ for $r\in{\overline}{1,n-1}$, $$i_n=q_{n,n}^0+q_{n,n}+q^1_{n,n}.\eqno(4.48)$$ In particular, (4.44) implies $$p_{r_2,r_1}=\sum_{s=1}^{r_1-1}p^0_{r_2,s}\qquad{\mbox{for}}\;\;2\leq r_1<r_2\leq n.\eqno(4.49)$$

Set $$\Im_r=\sum_{s=1}^nq_{s,r+1}+\sum_{s=2}^{r+1}q_{r+1,s}^1+\sum_{s=r+1}^nq_{s,r+1}^1\eqno(4.50)$$ for $r\in{\overline}{1,n-1}$. By (4.42) and (4.45), we have $$q_{1,2}+q^1_{2,2}=q_{1,1}^0.\eqno(4.51)$$ Moreover, (4.46) implies $$q_{i,2}+q_{1,i+1}+q_{i+1,2}^1=q_{1.i}+q_{i,1}^0\qquad{\mbox{for}}\;\;i\in{\overline}{2,n}.\eqno(4.52)$$ $$2(q_{1,2}+q^1_{2,2})+\sum_{i=2}^nq_{i,2}+\sum_{r=3}^n(q_{1,r}+q_{r,2}^1)=2q^0_{1,1}
+\sum_{i=2}^n(q_{1,i}+q^0_{i,1}),\eqno(4.53)$$ equivalently, $$\sum_{i=1}^nq_{i,2}+q^1_{2,2}+\sum_{r=2}^nq_{r,2}^1=q^0_{1,1}+\sum_{i=1}^nq^0_{i,1}.\eqno(4.54)$$ Thus $$\Im_1=q^0_{1,1}+\sum_{i=1}^nq^0_{i,1}.\eqno(4.55)$$

Let $r\in{\overline}{2,n-1}$. By (4.46), we have $$q_{i,r+1}+q_{r,i+1}+q_{i+1,r+1}^1=q_{r.i}+q_{i,r}+q_{i,r}^0+q_{i,r}^1\eqno(4.56)$$ for $i\in{\overline}{r+1,n}$ and $$q_{r,j+1}+q_{j,r+1}+q_{r+1,j+1}^1=q_{j.r}+q_{r,j}+q_{r,j}^0+q_{r,j}^1\eqno(4.57)$$ for $j\in{\overline}{1,r-1}$. Expressions (4.45), (4.56) and (4.57) imply $$\begin{aligned}
 & &2(q_{r,r+1}+q_{r+1,r+1}^1)+\sum_{i=r+1}^n(q_{i,r+1}+q_{r,i+1}
+q_{i+1,r+1}^1)+\sum_{j=1}^{r-1}(q_{r,j+1}+q_{j,r+1}+q_{r+1,j+1}^1)\\
&=& 2(q^0_{r,r}+q_{r,r}++q_{r,r}^1)
+q_{i,r}^1)+\sum_{j=1}^{r-1}(q_{j.r}+q_{r,j}+q_{r,j}^0+q_{r,j}^1)\\&
&
+\sum_{s=r+1}^n(q_{r,s}+q_{s,r}+q_{s,r}^0+q_{s,r}^1),\hspace{8.4cm}(4.58)\end{aligned}$$ equivalently, $$\begin{aligned}
\qquad & &\sum_{i=1}^nq_{i,r+1}+\sum_{j=2}^{r+1}q_{r+1,j}^1+\sum_{i=r+1}^n q_{i,r+1}
\\ &=&\sum_{j=1}^rq_{r,j}^0+\sum_{s=r}^nq_{s,r}^0+\sum_{s=1}^nq_{s,r}+\sum_{j=2}^rq_{r,j}^1
+\sum_{s=r}^nq_{s,r}^1\hspace{4.9cm}(4.59)\end{aligned}$$ by (4.42). So $$\Im_r-\Im_{r-1}=\sum_{j=1}^rq_{r,j}^0+\sum_{i=r}^nq_{i,r}^0.\eqno(4.60)$$ Furthermore, by (4.47), (4.49), (4.50), (4.60) and induction, we have $$i_r=\sum_{s=r+1}^n\sum_{j=1}^rp_{s,j}^0+\sum_{s=1}^r(\sum_{j=1}^sq_{s,j}^0+\sum_{j=s}^nq_{j,s}^0)
\eqno(4.61)$$ for $r\in{\overline}{1,n-1}$. In particular, $$i_r-i_{r-1}=\sum_{s=r+1}^np^0_{s,r}-\sum_{s=1}^{r-1}p^0_{r,s}+\sum_{j=1}^rq_{r,j}^0
+\sum_{s=r}^nq_{s,r}^0\eqno(4.62)$$ for $r\in{\overline}{1,n-1}$.

Next, we want to calculate $i_n$. First, we have $$q_{n.n}+q^1_{n,n}=q_{n,n-1}+q_{n-1,n}+q_{n,n}^1+q_{n,n-1}^1+q_{n,n-1}^0\eqno(4.63)$$ by (4.46). Moreover, for $r\in{\overline}{1,n-2}$, we have $$\begin{aligned}
& &\sum_{i=0}^{2r}q_{n-i,n-2r+i}+\sum_{i=0}^{r-1}q_{n-i,n-2r+1+i}
\\ &=&q_{n,n-2r}+\sum_{i=1}^r(q_{n-i,n-2r+i}+q_{n-2r-1+i,n-i+1}+q_{n-i+1,n-2r+i}^1)\\ &=&
\sum_{i=1}^r(q_{n-i,n-2r-1+i}+q_{n-2r-1+i,n-i}+q_{n-i,n-2r-1+i}^0+q_{n-i,n-2r-1+i}^1)
\hspace{4cm}\end{aligned}$$ $$\begin{aligned}
 && +q_{n,n-2r-1}+q_{n-2r-1,n}+q_{n,n-2r-1}^0+q_{n,n-2r-1}^1
\\ &=& \sum_{i=0}^r(q_{n-i,n-2r-1+i}+q_{n-2r-1+i,n-i})+\sum_{i=0}^rq_{n-i,n-2r-1+i}^1+
 \sum_{i=0}^rq_{n-i,n-2r-1+i}^0\\ &=&\sum_{i=0}^{2r+1}q_{n-i,n-2r-1+i}+ \sum_{i=0}^r
 q_{n-i,n-2r-1+i}^0+ \sum_{i=0}^rq_{n-i,n-2r-1+i}^1.\hspace{3.7cm}(4.64)\end{aligned}$$ by (4.46). Furthermore, for $r\in{\overline}{0,n-2}$, we get $$\begin{aligned}
& &\sum_{i=0}^{2r+1}q_{n-i,n-2r-1+i}+\sum_{i=0}^rq_{n-i,n-2r+i}\\
&=&q_{n,n-2r-1}+q_{n-r-1,n-r}+q_{n-r,n-r}^1+\sum_{i=1}^r(q_{n-i,n-2r-1+i}\\
& & +q_{n-2r-2+i,n+1-i} +q_{n-i+1,n-2r-1+i}^1)
\\ &=& q_{n,n-2r-2}+q_{n-2r-2,n}+q^0_{n,n-2r-2}+q^1_{n,n-2r-2}+q_{n-r-1,n-r-1}
+q_{n-r-1,n-r-1}^0\\ & &+q_{n-r-1,n-r-1}^1
+\sum_{i=1}^r(q_{n-i,n-2r-2+i}+q_{n-2r-2+i,n-i}+q_{n-i,n-2r-2+i}^0+q_{n-i,n-2r-2+i}^1)
\\ &=&
q_{n-r-1,n-r-1}+\sum_{i=0}^r(q_{n-i,n-2r-2+i}+q_{n-2r-2+i,n-i})
\\ & &+\sum_{i=0}^{r+1}q_{n-i,n-2r-2+i}^0+\sum_{i=0}^{r+1}q_{n-i,n-2r-2+i}^1\\
&=& \sum_{i=0}^{2r+2}
q_{n-i,n-2r-2+i}+\sum_{i=0}^{r+1}q_{n-i,n-2r-2+i}^0+\sum_{i=0}^{r+1}q_{n-i,n-2r-2+i}^1
\hspace{3.9cm}(4.65)\end{aligned}$$ based on (4.45) and (4.46).

By (4.63)-(4.65) and induction, we obtain $$q_{n,n}+q_{n,n}^1=\sum_{r=0}^{n-1}\sum_{s=0}^rq_{n-s,n-2r-1+s}^0+
\sum_{r=0}^{n-2}\sum_{s=0}^rq_{n-s,n-2r-2+s}^0=\sum_{1\leq r_1\leq
r_2\leq n}q^0_{r_2,r_1} -q_{n,n}^0.\eqno(4.66)$$ Therefore, $$i_n=\sum_{1\leq r_1\leq r_2\leq n}q_{s_2,s_1}^0.\eqno(4.67)$$ Now $$2i_n-i_{n-1}=\sum_{1\leq r_1\leq r_2\leq n}2q_{s_2,s_1}^0-\sum_{r=1}^{n-1}(p^0_{n,r}
+\sum_{p=1}^rq_{r,i}^0+\sum_{s=r}^nq_{s,r}^0)=q_{n,n}^0+\sum_{r=1}^nq_{n,r}^0
-\sum_{s=1}^{n-1}p^0_{n,s}.\eqno(4.68)$$

By (4.45) and induction on $r$, we get $$q_{r,r}^1=\sum_{s=1}^{r-1}(q_{s,s}^0+q_{s,s}-q_{s,s+1})\qquad{\mbox{for}}\;\;r\in{\overline}{2,n}.\eqno(4.69)$$ Moreover, (4.46) and induction imply $$q_{n,r}=\sum_{s=1}^{r-1}(q^0_{n,s}+q_{s,n}+q_{n,s}^1)\qquad{\mbox{for}}\;\;r\in{\overline}{2,n}\eqno(4.70)$$ and $$q_{r_2,r_1}^1=\sum_{s=1}^{r_2-1}(q_{r_2-s,r_1-s}^0+q_{r_2-s,r_1-s}+q_{r_1-s,r_2-s})
-\sum_{s=1}^{r_1-1}(q_{r_2-s,r_1-s+1}+q_{r_1-s,r_2-s+1})\eqno(4.71)$$ for $2\leq r_1<r_2\leq n$. Furthermore, (4.70) and (4.71) show $$\begin{aligned}
\qquad q_{n,r}&=&\sum_{s=1}^{r-1}(q_{n,i}^0+q_{i,n})+\sum_{s=2}^{r-1}\sum_{j=1}^{s-1}(q_{n-j,s-j}^0+
q_{n-j,s-j}+q_{s-j,n-j})\\ &
&-\sum_{s=2}^{r-1}\sum_{j=1}^{s-1}(q_{n-j,s-j+1}+q_{s-j,n-j+1})\hspace{6.4cm}
(4.72)\end{aligned}$$ for $r\in{\overline}{2,n}$. Set $$\vec{p^0}!=\prod_{1\leq r_1<r_2\leq n}p_{r_2,r_1}^0!,\qquad \vec{q^0}!=
\prod_{1\leq r_1\leq r_2\leq n}q_{r_2,r_1}^0!.\eqno(4.73)$$

By (4.40), (4.43), (4.61), (4.67), (4.69), (4.71) and (4.72), we have $$\begin{aligned}
& &E_C(z_1,z_2,...,z_n)\\ &=& \sum_{{\alpha}\in
{\Gamma}_C}\; \sum_{1\leq j_1<j_2\leq
n}\;\sum_{k_1=1}^n\;\sum_{k_2=2}^n\;\sum_{k_3=k_1}^n\;\sum_{k_4=1}^{n-1}\;
\sum_{p_{j_2,j_1}^0,
q_{k_3,k_1}^0,q_{k_4,k_2}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}}{i_1!i_2!\cdots
i_n! 2^{i_n+\sum_{r=1}^nq^0_{r,r}}{\langle}{\lambda}_n{\rangle}_{i_n}}
\\ & &\times \left[\prod_{2\leq l_1\leq l_2\leq n}\left(\!\!\begin{array}{c}{\alpha}_{l_2,l_1}\\
p^0_{l_2,l_1},
p_{l_2,l_1}\end{array}\!\!\right)\left(\!\!\begin{array}{c}{\alpha}_{n+l_2,l_1}\\
q^0_{l_2,l_1},q_{l_2,l_1}^1,q_{l_1,l_2},q_{l_2,l_1}\end{array}\!\!\right)\right]\\
& &\times\left[\prod_{m=2}^n\left(\!\!\begin{array}{c}{\alpha}_{m,1}\\
p^0_{m,1}\end{array}\!\!\right)\right]\left(\!\!\begin{array}{c}{\alpha}_{n+1,1}\\
q^0_{1,1}\end{array}\!\!\right)\left[\prod_{r=2}^n\left(\!\!\begin{array}{c}{\alpha}_{n+r,r}\\
q^0_{r,r},q_{r,r}^1,q_{r,r}\end{array}\!\!\right)\left(\!\!\begin{array}{c}{\alpha}_{n+r,1}\\
q^0_{r,1},q_{1,r}\end{array}\!\!\right)\right]\\ & &\times
\left[\prod_{s=1}^{n-2}{\langle}i_s{\rangle}_{_{\sum_{r_1=s+2}^np_{r_1,s+1}+\sum_{r=1}^nq_{r,s+1}+\sum_{r=2}^{s+1}q_{s+1,r}^1
+\sum_{r=s+1}^nq_{r,s+1}^1}}\right] {\langle}i_{n-1}{\rangle}_{_{\sum_{r=1}^nq_{r,n}+\sum_{r=2}^nq_{n,r}^1+q^1_{n,n}}}\\
& &\times \left[\prod_{s=1}^{n-1}{\langle}i_s
-i_{s-1}-1/2{\rangle}_{_{\sum_{r=s+1}^np_{r,s}^0+\sum_{r=1}^sq_{s,r}^0+\sum_{r=s}^nq_{r,s}^0}}\right]{\langle}2i_n-i_{n-1}-1/2{\rangle}_{_{2q^0_{n,n}+\sum_{r=1}^{n-1}q^0_{n,r}}}\\
& &\times\left[\prod_{1\leq r_1<r_2\leq
n}\left(\frac{z_{r_2}}{z_{r_1}}\right)^{{\alpha}_{r_2,r_1}}\frac{1}{(z_{r_1}z_{r_2})
^{{\alpha}_{n+r_2,r_1}}}\right]\left[\prod_{r=1}^n\frac{1}{z_r^{{\alpha}_{n+r,r}}}\right](z_1z_2\cdots
z_{n-1})^{-1/2}z_n^{{\lambda}_n}\\ &=&\sum_{1\leq j_1<j_2\leq
n}\;\sum_{k_1=1}^n\;\sum_{k_2=2}^n\;\sum_{k_3=k_1}^n\;\sum_{k_4=1}^{n-1}
\;\sum_{p_{j_2,j_1}^0,
q_{k_3,k_1}^0,q_{k_4,k_2}=0}^{\infty}\frac{(-1)^{i_1+i_2+\cdots+i_{n-1}}}{\vec
p^0! 2^{i_n+\sum_{r=1}^nq^0_{r,r}}{\langle}{\lambda}_n{\rangle}_{i_n}}\\ &
&\times\frac{1}{q_{n,n}^0!q_{n,n}!q_{n,n}^1!} \left[\prod_{1\leq
l_1\leq l_2\leq n}
\left(\!\!\begin{array}{c}q^0_{l_2,l_1}+q_{l_2,l_1}^1+q_{l_1,l_2}+q_{l_2,l_1}
\\ q_{l_2,l_1}^1,q_{l_1,l_2},q_{l_2,l_1}\end{array}\!\!\right)\right]\left[\prod_{r=2}^{n-1}
\left(\!\!\begin{array}{c}q^0_{r,r}+q_{r,r}^1+q_{r,r}\\
q_{r,r}^1,q_{r,r}\end{array}\!\!\right)\right]\end{aligned}$$ $$\begin{aligned}
 & &\times
\left[\prod_{s=1}^{n-1}{\langle}i_s-i_{s-1}-1/2{\rangle}_{_{\sum_{r=s+1}^np_{r,s}^0+\sum_{r=1}^sq_{s,r}^0+\sum_{r=s}^nq_{r,s}^0}}
\right]{\langle}2i_n-i_{n-1}-1/2{\rangle}_{_{2q^0_{n,n}+\sum_{r=1}^{n-1}q^0_{n,r}}}\\
& &\times \left[\prod_{1\leq r_1<r_2\leq
n}\frac{z_{r_1}^2z_{r_2}^{1+\sum_{s=1}^{r_1}p_{r_2,s}^0}}{(z_{r_1}-z_{r_2})
^{1+\sum_{s=1}^{r_1}p_{r_2,s}^0}(z_{r_1}z_{r_2}-1)^{1+q^0_{r_2,r_1}+q_{r_2,r_1}^1
+q_{r_1,r_2}+q_{r_2,r_1}}}\right]\\ & &\times
\left[\prod_{r=1}^n\frac{z_r^2}{(z_r^2-1)^{1+q^0_{r,r}+q^1_{r,r}+q_{r,r}}}\right]
(z_1z_2\cdots
z_{n-1})^{-1/2}z_n^{{\lambda}_n}.\hspace{5cm}(4.74)\end{aligned}$$ Note that $$\begin{aligned}
& &\frac{1}{q_{n,n}^0!q_{n,n}!q_{n,n}^1!} \left[\prod_{1\leq
l_1\leq l_2\leq n}
\left(\!\!\begin{array}{c}q^0_{l_2,l_1}+q_{l_2,l_1}^1+q_{l_1,l_2}+q_{l_2,l_1}
\\ q_{l_2,l_1}^1,q_{l_1,l_2},q_{l_2,l_1}\end{array}\!\!\right)\right]\left[\prod_{r=2}^{n-1}
\left(\!\!\begin{array}{c}q^0_{r,r}+q_{r,r}^1+q_{r,r}\\
q_{r,r}^1,q_{r,r}\end{array}\!\!\right)\right]\\
&=&\frac{1}{q_{n,n}^0!q_{n,n}!q_{n,n}^1!} \left[\prod_{1\leq
l_1\leq l_2\leq n}
\left(\!\!\begin{array}{c}q_{l_2+1,l_1+1}^1+q_{l_1,l_2+1}+q_{l_2,l_1+1}
\\ q_{l_2,l_1}^1,q_{l_1,l_2},q_{l_2,l_1}\end{array}\!\!\right)\right]\left[\prod_{r=2}^{n-1}
\left(\!\!\begin{array}{c}q_{r+1,r+1}^1+q_{r,r+1}\\
q_{r,r}^1,q_{r,r}\end{array}\!\!\right)\right]\\ &=&\frac{1}{\vec
q^0!} \left[\prod_{1\leq l_1\leq l_2\leq n-1}
\left(\!\!\begin{array}{c}q_{l_2+1,l_1+1}^1+q_{l_1,l_2+1}+q_{l_2,l_1+1}
\\ q_{l_1,l_2+1},q_{l_2,l_1+1}\end{array}\!\!\right)\right]\left[\prod_{r=1}^{n-1}
\left(\!\!\begin{array}{c}q_{r+1,r+1}^1+q_{r,r+1}\\
q_{r,r+1}\end{array}\!\!\right)\right]\\ &=&\frac{1}{\vec q^0!}
\left[\prod_{1\leq l_1\leq l_2\leq n-1}
\left(\!\!\begin{array}{c}q^0_{l_2,l_1}+q_{l_2,l_1}^1+q_{l_1,l_2}+q_{l_2,l_1}
\\ q_{l_1,l_2+1},q_{l_2,l_1+1}\end{array}\!\!\right)\right]\left[\prod_{r=1}^{n-1}
\left(\!\!\begin{array}{c}q^0_{r,r}+q_{r,r}^1+q_{r,r}\\
q_{r,r+1}\end{array}\!\!\right)\right]\hspace{0.9cm}(4.75)\end{aligned}$$ according to (4.45) and (4.46). Furthermore, $$q_{r,r}^0+q^1_{r,r}+q_{r,r}=\sum_{s=1}^r(q_{s,s}^0+q_{s,s})-\sum_{s=1}^{r-1}q_{s,s+1}
\eqno(4.76)$$ by (4.69) and $$\begin{aligned}
& & q_{n,r}^0+q_{n,r}^1+q_{r,n}+q_{n,r}\\
&=&\sum_{s=1}^r(q^0_{n,s}+q_{s,n})+\sum_{s=0}^{r-2}\sum_{j=1}^{r-s-1}(q^0_{n-j,r-s-j}
+q_{n-j,r-s-j}+q_{r-s-j,n-j})\\ &
&-\sum_{s=0}^{r-2}\sum_{j=1}^{r-s-1}(q_{n-j,r-s-j+1}+q_{r-s-j,n-j+1})\\
&=&\sum_{s=0}^{r-1}\sum_{j=0}^{r-s-1}q^0_{n-j,r-s-j}+\sum_{j=0}^{r-1}q_{r-j,n-j}-\sum_{j=1}^{r-1}
q_{r-j,n-j+1}\hspace{5cm}(4.77)\end{aligned}$$ by (4.70) and (4.72) for $r\in{\overline}{1,n-1}$, and $$\begin{aligned}
& &
q_{r_2,r_1}^0+q_{r_2,r_1}^1+q_{r_1,r_2}+q_{r_2,r_1}\\
&=&\sum_{s=0}^{r_1-1}(q_{r_2-s,r_1-s}^0+q_{r_2-s,r_1-s}+q_{r_1-s,r_2-s})
-\sum_{s=1}^{r_1-1}(q_{r_2-s,r_1-s+1}+q_{r_1-s,r_2-s+1})
\hspace{1.2cm}(4.78)\end{aligned}$$ by (4.71) for $1\leq r_1<
r_2\leq n-1.$

Let $$y_{r_2,r_1}=\frac{1}{z_{r_1}z_{r_2}-1}\qquad{\mbox{for}}\;\;1\leq
r_1\leq r_2\leq n.\eqno(4.79)$$ Set $$w_{r_2,r_1}=\frac{y_{r_2,r_1}y_{r_2+1,r_1+1}\cdots
y_{n-1,n+r_1-r_2-1}}{y_{r_2+1,r_1}y_{r_2+2,r_1+1}\cdots
y_{n,n+r_1-r_2-1}}\qquad{\mbox{for}}\;\;2\leq r_1\leq r_2\leq
n-1,\eqno(4.80)$$ $$w_{k,n}=y_{n,k}\qquad{\mbox{for}}\;\;k\in{\overline}{1,n-1}\eqno(4.81)$$ and $$w_{r_1,r_2}=\frac{y_{r_2,r_1}y_{r_2+1,r_1+1}\cdots
y_{n,n+r_1-r_2}}{y_{r_2,r_1+1}y_{r_2+1,r_1+2}\cdots
y_{n-1,n+r_1-r_2}}\qquad{\mbox{for}}\;\;1\leq r_1<r_2\leq
n-1.\eqno(4.82)$$ Based on (4.76)-(4.78), we have $$\begin{aligned}
& &\prod_{1\leq r_1\leq r_2\leq
n}y_{r_2,r_1}^{q^0_{r_2,r_1}+q^1_{r_2,r_1}+q_{r_2,r_1}+q_{r_1,r_2}}=\left[\prod_{r_1=1}^{n-1}
\prod_{r_2=2}^nw_{r_1,r_2}^{q_{r_1,r_2}}\right]\\ &&\times
\left[\prod_{r=1}^n(y_{r,r}y_{r+1,r+1}\cdots
y_{n,n})^{q_{r,r}^0}\right]\left[\prod_{r=1}^{n-1}(y_{n,r}y_{n,r+1}\cdots
y_{n,n})^{q_{n,r}^0}\right]\prod_{1\leq r_1<r_2\leq n-1}\\ &
&(y_{r_2,r_1}y_{r_2+1,r_1+1}\cdots
y_{n-1,n+r_1-r_2-1}y_{n,n+r_1-r_1}y_{n,n+r_1-r_1+1}\cdots
y_{n,n})^{q^0_{r_2,r_1}}.\hspace{2.4cm}(4.83)\end{aligned}$$

Next we have $$\begin{aligned}
\hspace{2cm}& &\sum_{q_{n-1,n}=0}^{\infty}
\left(\!\!\begin{array}{c}q^0_{n-1,n-1}+q_{n-1,n-1}^1+q_{n-1,n-1}\\
q_{n-1,n}\end{array}\!\!\right)w_{n-1,n}^{q_{n-1,n}}\\
&=&(1+w_{n-1,n})^{q^0_{n-1,n-1}+q_{n-1,n-1}^1+q_{n-1,n-1}}\\
&=&(1+w_{n-1,n})^{\sum_{s=1}^{n-1}(q_{s,s}^0+q_{s,s})-\sum_{s=1}^{n-2}q_{s,s+1}}
\hspace{5.5cm}(4.84)\end{aligned}$$ by (4.76). Set $$w^{(2n-1)}_{r,r+1}=\frac{w_{r,r+1}}{1+w_{n-1,n}},\qquad
w_{s,s}^{(2n-1)}=(1+w_{n-1,n})w_{s,s}\eqno(4.85)$$ for $r\in{\overline}{1,n-2},\;s\in{\overline}{1,n-1}$ and $$w^{(2n-1)}_{r_1,r_2}=w_{r_1,r_2}\qquad{\mbox{for}}\;\;r_1\in{\overline}{1,n-1},\;r_1,r_1+1\neq
r_2\in{\overline}{2,n}.\eqno(4.86)$$ Suppose that we have defined $\{w_{r_1,r_2}^{(2k+1)}\mid r_1+r_2<2k+1\}$. Set $$\ell=\max\{1,2k-n\}.\eqno(4.87)$$ Note $$\begin{aligned}
&&
\sum_{q_{\ell,2k-\ell},q_{\ell+1,2k-\ell-1},\cdots,q_{2k-\ell-1,\ell+1}=0}^{\infty}\;
\prod_{r=\ell}^{k-1}\left(\!\!\begin{array}{c}q^0_{2k-1-r,r}+q^1_{2k-1-r,r}+q_{2k-1-r,r}+q_{r,2k-1-r}\\
q_{r,2k-r},q_{2k-r-1,r+1}\end{array}\!\!\right)\hspace{2cm}\end{aligned}$$ $$\begin{aligned}
 & &\times
w_{r,2k-r}^{(2k+1)q_{r,2k-r}}w_{2k-r-1,r+1}^{(2k+1)q_{2k-r-1,r+1}}\\
&=&\prod_{r=\ell}^{k-1}(1+w_{r,2k-r}^{(2k+1)}+w_{2k-r-1,r+1}^{(2k+1)})
^{q^0_{2k-1-r,r}+q^1_{2k-1-r,r}+q_{2k-1-r,r}+q_{r,2k-1-r}}\\
&=&\prod_{r=\ell}^{k-1}(1+w_{r,2k-r}^{(2k+1)}+w_{2k-r-1,r+1}^{(2k+1)})^{\sum_{s=1}^r
(q^0_{2k-1-2r+s,s}+q_{2k-1-2r+s,s}+q_{s,2k-1-2r+s})}\\ &
&\times(1+w_{r,2k-r}^{(2k+1)}+w_{2k-r-1,r+1}^{(2k+1)})^{-\sum_{s=1}^r
(q_{2k-1-2r+s,s+1}+q_{s,2k-2r+s})}\hspace{3.7cm}(4.88)\end{aligned}$$ by (4.78). Set $$w_{2k-1-2r+s,s}^{(2k)}=w_{2k-1-2r+s,s}^{(2k+1)}(1+w_{r,2k-r}^{(2k+1)}+w_{2k-r-1,r+1}^{(2k+1)}),
\eqno(4.89)$$ $$w_{s,2k-1-2r+s}^{(2k)}=w_{s,2k-1-2r+s}^{(2k+1)}(1+w_{r,2k-r}^{(2k+1)}+w_{2k-r-1,r+1}^{(2k+1)})
\eqno(4.90)$$ for $1\leq s\leq r\leq k-1$; $$w_{2k-1-2r+s,s+1}^{(2k)}=\frac{w_{2k-1-2r+s,s+1}^{(2k+1)}}{1+w_{r,2k-r}^{(2k+1)}
+w_{2k-r-1,r+1}^{(2k+1)}}, \eqno(4.91)$$ $$w_{s,2k-2r+s}^{(2k)}=\frac{w_{s,2k-2r+s}^{(2k+1)}}{1+w_{r,2k-r}^{(2k+1)}+
w_{2k-r-1,r+1}^{(2k+1)}} \eqno(4.92)$$ for $1\leq s\leq r\leq k-1$ and $$w^{(2k)}_{r_1,r_2}=w^{(2k+1)}_{r_1,r_2}\;\;\mbox{for the other
pairs}\;\;(r_1,r_2)\;\;\mbox{such that}\;\;r_1+r_2\leq
2k.\eqno(4.93)$$

Let $$\iota=\max\{1,2k-1-n\}.\eqno(4.94)$$ Moreover, $$\begin{aligned}
&&\sum_{q_{\iota,2k-\iota-1},q_{\iota+1,2k-\iota-2},\cdots,
q_{2k-\iota-2,\iota+1}=0}
^{\infty}[\prod_{r=\iota}^{k-2}\left(\!\!\begin{array}{c}q^0_{2k-2-r,r}+q^1_{2k-2-r,r}+q_{2k-2-r,r}+q_{r,2k-2-r}
\\ q_{r,2k-r-1},q_{2k-r-2,r+1}\end{array}\!\!\right)\\ &&\times
w_{r,2k-r-1}^{(2k)q_{r,2k-r-1}}w_{2k-r-2,r+1}^{(2k)q_{2k-r-2,r+1}}]\left(\!\!\begin{array}{c}
q^0_{k-1,k-1}+q^1_{k-1,k-1}
+q_{k-1,k-1}\\ q_{k-1,k}\end{array}\!\!\right)w_{k-1,k}^{(2k)q_{k-1,k}}\\
&=&\left[\prod_{r=\iota}^{k-2}(1+w_{r,2k-r-1}^{(2k)}+w^{(2k)}_{2k-r-2,r+1})
^{q^0_{2k-2-r,r}+q^1_{2k-2-r,r}+q_{2k-2-r,r}+q_{r,2k-2-r}}\right]\\
&&\times (1+w_{k-1,k}^{(2k)})^{q^0_{k-1,k-1}+q^1_{k-1,k-1}
+q_{k-1,k-1}}\\
&=&[\prod_{r=\iota}^{k-2}(1+w_{r,2k-r-1}^{(2k)}+w^{(2k)}_{2k-r-2,r+1})
^{\sum_{s=1}^r(q^0_{2k-2-2r+s,s}+q_{2k-2-2r+s,s}+q_{s,2k-2-2r+s})}\\
& &\times(1+w_{r,2k-r-1}^{(2k)}+w^{(2k)}_{2k-r-2,r+1})
^{-\sum_{s=1}^{r-1}(q_{2k-2-2r+s,s+1}+q_{s,2k-1-2r+s})}\\ &&\times
(1+w_{k-1,k}^{(2k)})^{\sum_{r=1}^{k-1}(q^0_{r,r}+q_{r,r})-\sum_{r=1}^{k-1}q_{r,r+1}}
\hspace{7cm}(4.95)\end{aligned}$$ by (4.76) and (4.78). Set $$w_{2k-2-2r+s,s}^{(2k-1)}=w_{2k-2-2r+s,s}^{(2k)}(1+w_{r,2k-r-1}^{(2k)}+w_{2k-r-2,r+1}^{(2k)}),
\eqno(4.96)$$ $$w_{s,2k-2-2r+s}^{(2k-1)}=w_{s,2k-2-2r+s}^{(2k)}(1+w_{r,2k-r-1}^{(2k)}+w_{2k-r-2,r+1}^{(2k)})
\eqno(4.97)$$ for $1\leq s\leq r\leq k-2$; $$w_{2k-2-2r+s,s+1}^{(2k-1)}=\frac{w_{2k-2-2r+s,s+1}^{(2k)}}{1+w_{r,2k-r-1}^{(2k)}
+w_{2k-r-2,r+1}^{(2k)}}, \eqno(4.98)$$ $$w_{s,2k-1-2r+s}^{(2k-1)}=\frac{w_{s,2k-1-2r+s}^{(2k)}}{1+w_{r,2k-r-1}^{(2k)}+
w_{2k-r-2,r+1}^{(2k)}} \eqno(4.99)$$ for $1\leq s\leq r\leq k-2$; $$w_{r,r}^{(2k-1)}=w_{r,r}^{(2k)}(1+w_{k-1,k}^{(2k)}),\qquad
w_{s,s+1}^{(2k-1)}=\frac{w_{s,s+1}^{(2k)}}{1+w_{k-1,k}^{(2k)}}\eqno(4.100)$$ for $r\in{\overline}{1,k-1},\;s\in{\overline}{1,k-2}$; and $$w^{(2k-1)}_{r_1,r_2}=w^{(2k)}_{r_1,r_2}\;\;\mbox{for the other
pairs}\;\;(r_1,r_2)\;\;\mbox{such that}\;\;r_1+r_2\leq
2k-1.\eqno(4.101)$$ By induction, we have defined $$\{w^{(k)}_{r_1,r_2}\mid
k\in{\overline}{1,2n-1};\;r_1\in{\overline}{1,n-1},\;r_2\in{\overline}{1,n},\;r_1+r_2\leq
k\}.\eqno(4.102)$$ For convenience, we denote $$w_{r_1,r_2}^{(2n)}=w_{r_1,r_2}\qquad{\mbox{for}}\;\;r_1\in{\overline}{1,n-1},\;r_2\in{\overline}{2,n}.\eqno(4.103)$$ Based on (4.84), (4.88) and (4.95), we have $$\begin{aligned}
&&\sum_{r_1=1}^{n-1}\;\sum_{r_2=2}^n\;\sum_{q_{r_1,r_2}=0}^{\infty}
[\prod_{1\leq l_1\leq l_2\leq n-1}
\left(\!\!\begin{array}{c}q^0_{l_2,l_1}+q_{l_2,l_1}^1+q_{l_1,l_2}+q_{l_2,l_1}
\\ q_{l_1,l_2+1},q_{l_2,l_1+1}\end{array}\!\!\right)\\ & &\times
w_{l_1,l_2+1}^{q_{l_1,l_2+1}}w_{l_2,l_1+1}^{q_{l_2,l_1+1}}]\left[\prod_{r=1}^{n-1}
\left(\!\!\begin{array}{c}q^0_{r,r}+q_{r,r}^1+q_{r,r}\\
q_{r,r+1}\end{array}\!\!\right)w_{r,r+1}^{q_{r,r+1}}\right]\\
&=&\left[\prod_{k=2}^n(1+w_{k-1,k}^{(2k)})^{\sum_{r=1}^{k-1}q_{r,r}^0}\right]\left[\prod_{k=2}
^{n-1}\prod_{r=\ell}^{k-1}(1+w_{r,2k-r}^{(2k+1)}+w_{2k-r-1,r+1}^{(2k+1)})^{\sum_{s=1}^r
q_{2k-1-2r+s,s}^0}\right]\\ &&\times
\left[\prod_{k=3}^{n-1}\prod_{r=\iota}^{k-2}(1+w_{r,2k-r-1}^{(2k)}+w_{2k-r-2,r+1}^{(2k)})
^{\sum_{s=1}^rq_{2k-2-2r+s,s}^0}\right]\\ &=&[\prod_{1\leq
r_1<r_2\leq n-1;r_2-r_1\;\mbox{\small is
odd}}\;(\prod_{k=(r_2-r_1+3)/2}^{n-1}(1+w^{(2k+1)}_{(2k+r_1-r_2-1)/2,(2k-r_1+r_2+1)/2}\\
&&+w^{(2k+1)}_{(2k-r_1+r_2-1)/2,(2k+r_1-r_2+1)/2})^{q^0_{r_2,r_1}})]
[\prod_{1\leq r_1<r_2\leq n-1;r_2-r_1\;\mbox{\small is
even}}\hspace{4cm}\end{aligned}$$ $$\begin{aligned}
& &
(\prod_{k=(r_2-r_1+2)/2}^{n-1}(1+w^{(2k+2)}_{(2k+r_1-r_2)/2,(2k-r_1+r_2+2)/2}+w^{(2k+2)}
_{(2k-r_1+r_2)/2,(2k+r_1-r_2+2)/2})^{q^0_{r_2,r_1}})]\\ & &\times
\left[\prod_{r=1}^{n-1}\left(\prod_{k=r}^{n-1}(1+w_{k,k+1}^{(2k)})\right)^{q_{r,r}^0}\right].
\hspace{8.4cm}(4.104)\end{aligned}$$

According to (4.74), (4.83) and (4.104), we define $$\xi_{n,i}^C=(-1)^{n+i+1}\frac{y_{n,i}y_{n,i+1}\cdots
y_{n,n}}{2^{1+{\delta}_{n,i}}}\qquad{\mbox{for}}\;\;1\leq i\leq
n,\eqno(4.105)$$ $$\xi_{i,i}^C=-\frac{y_{i,i}\cdots
y_{n,n}\prod_{k=i}^{n-1}(1+w_{k,k+1}^{(2k)})}{4}\eqno(4.106)$$ for $1\leq i\leq n-1$, $$\begin{aligned}
&
&\xi_{r_2,r_1}^C=\frac{(-1)^{r_1+r_2+1}}{2}y_{r_2,r_1}y_{r_2+1,r_1+1}\cdots
y_{n-1,n+r_1-r_2-1}y_{n,n+r_1-r_2}y_{n,n+r_1-r_2+1}\cdots
y_{n,n}\\ &
&\prod_{k=(r_2-r_1+3)/2}^{n-1}(1+w^{(2k+1)}_{(2k+r_1-r_2-1)/2,(2k-r_1+r_2+1)/2}
+w^{(2k+1)}_{(2k-r_1+r_2-1)/2,(2k+r_1-r_2+1)/2})\hspace{0.5cm}(4.107)\end{aligned}$$ for $1\leq r_1<r_2\leq n-1$ such that $r_2-r_1$ is odd, and $$\begin{aligned}
&
&\xi_{r_2,r_1}^C=\frac{(-1)^{r_1+r_2+1}}{2}y_{r_2,r_1}y_{r_2+1,r_1+1}\cdots
y_{n-1,n+r_1-r_2-1}y_{n,n+r_1-r_2}y_{n,n+r_1-r_2+1}\cdots
y_{n,n}\\ &
&\prod_{k=(r_2-r_1+2)/2}^{n-1}(1+w^{(2k+2)}_{(2k+r_1-r_2)/2,(2k-r_1+r_2+2)/2}+w^{(2k+2)}
_{(2k-r_1+r_2)/2,(2k+r_1-r_2+2)/2})\hspace{1.2cm}(4.108)\end{aligned}$$ for $1\leq r_1<r_2\leq n-1$ such that $r_2-r_1$ is even.

For ${\alpha}\in{\Gamma}_C$, we set $${\alpha}^c_i=\sum_{r=1}^i{\alpha}_{n+r,i}+\sum_{s=i}^n{\alpha}_{n+s,i}\qquad{\mbox{for}}\;\;i\in{\overline}{1,n}\eqno(4.109)$$ and $${\alpha}^c=\sum_{1\leq r_1\leq r_2\leq n}{\alpha}_{n+r_2,r_1}.\eqno(4.110)$$

By (3.40), (4.74), (4.75), (4.79)-(4.83) and (4.104)-(4.110), we have $$\begin{aligned}
& &E_C(z_1,z_2,...,z_n)\\ &=&\frac{z_n^{n+{\lambda}_n+1}\prod_{i=1}^{n-1}
z_i^{2n-i+1/2}}{\left[\prod_{1\leq r_1<r_2\leq
n}(z_{r_1}-z_{r_2})\right]\left[\prod_{1\leq s_1 \leq s_2\leq
n}(z_{s_1}z_{s_2}-1)\right]}\sum_{{\gamma}\in{\Gamma}_C}\frac{1}{{\gamma}!{\langle}{\lambda}_n{\rangle}_{_{{\gamma}^c}}}
\\& &\times\prod_{r=1}^n{\langle}{\gamma}_{{\overline}{r}}-{\gamma}_{{\underline}{r}}+{\gamma}_r^c-1/2{\rangle}_{_{{\gamma}_{{\overline}{r}}+{\gamma}_r^c}}
\xi^{\gamma},\hspace{7.6cm}(4.111)\end{aligned}$$ where $${\gamma}!=\left[\prod_{1\leq r_1<r_2\leq n}{\gamma}_{r_2,r_1}!\right]\left[\prod_{1\leq s_1\leq s_2
\leq n}{\gamma}_{n+s_2,s_1}!\right]\eqno(4.112)$$ and $$\xi^{\gamma}=
\left[\prod_{1\leq r_1<r_2\leq
n}(\xi^A_{r_2,r_1})^{{\gamma}_{r_2,r_1}}\right] \left[\prod_{1\leq
s_1\leq s_2\leq
n}(\xi^C_{n+s_2,s_1})^{{\gamma}_{n+s_2,s_1}}\right].\eqno(4.112)$$ Recall the notations in (1.16) and (3.38). We define our hypergeometric function of type C by $${\cal X}_C(\tau_1,...,\tau_n;{\vartheta})\{z_{r_2,r_1},z_{n+s_2,s_1}\}=
\sum_{{\alpha}\in{\Gamma}_C}\frac{\prod_{r=1}^n(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}
{{\alpha}!({\vartheta})_{{\alpha}^c}}z^{\alpha}.\eqno(4.114)$$ By (2.18), (4.110) and (4.114), we obtain the following main theorem in this section:

<span>**Theorem 4.1**</span>. <span>*The Etingof trace function*</span> $$\begin{aligned}
E_C(z_1,z_2,...,z_n)&=&
\frac{z_n^{n+{\lambda}_n+1}\prod_{i=1}^{n-1}z_i^{2n-i+1/2}}
{\left[\prod_{1\leq r_1<r_2\leq
n}(z_{r_1}-z_{r_2})\right]\left[\prod_{1\leq s_1\leq s_2\leq
n}(z_{s_1}z_{s_2}-1)\right]}\\ & &\times {\cal
X}_C(1/2,...,1/2;-{\lambda}_n)\{\xi^A_{r_2,r_1},\xi^C_{n+s_2,s_1}\}.
\hspace{4cm}(4.115)\end{aligned}$$

Path Hypergeometric Functions
=============================

In this section, we find the differential properties, integral representations and differential equations for the hypergeometric functions of type A in (3.38), and the differential properties and differential equations for the hypergeometric functions of type C in (4.114). Moreover, we define our hypergeometric functions of type B and D analogously as those of type C.

For two positive integers $k_1$ and $k_2$ such that $k_1<k_2$, a <span>*path*</span> from $k_1$ to $k_2$ is a sequence $(m_0,m_1....,m_r)$ of positive integers such that $$k_1=m_0<m_1<m_2<\cdots <m_{r-1}<m_r=k_2.\eqno(5.1)$$ One can imagine a path from $k_1$ to $k_2$ is a way of a super man going from $k_1$th floor to $k_2$th floor through a stairway. Let $${\cal P}_{k_1}^{k_2}=\mbox{the set of all paths from}\;k_1\;\mbox{to}\;k_2.\eqno(5.2)$$ The <span>*path polynomial*</span> from $k_1$ to $k_2$ is defined as $$P_{[k_1,k_2]}=\sum_{(m_0,m_1,...,m_r)\in {\cal P}_{k_1}^{k_2}}(-1)^rz_{m_1,m_0}z_{m_2,m_1}
\cdots z_{m_{r-1},m_{r-2}}z_{m_r,m_{r-1}}.\eqno(5.3)$$ Moreover, we set $$P_{[k,k]}=1\qquad{\mbox{for}}\;\;0<k\in{\mathbb}{N}.\eqno(5.4)$$ For convenience, we simply denote $${\cal X}_A={\cal X}_A(\tau_1,..,\tau_n;{\vartheta})\{z_{j,k}\},\eqno(5.5)$$ $${\cal X}_A[i,j]={\cal X}_A(\tau_1,...,\tau_i+1,...,\tau_j-1,...\tau_n;{\vartheta})\{z_{r_2,r_1}\}
\eqno(5.6)$$ obtained from ${\cal X}_A$ by changing $\tau_i$ to $\tau_i+1$ and $\tau_j$ to $\tau_j-1$ for $1\leq i<j\leq n-1$ and $${\cal X}_A[k,n]={\cal X}_A(\tau_1,..,\tau_k+1,...,\tau_n+1;{\vartheta}+1)\{z_{r_2,r_1}\}\eqno(5.7)$$ obtained from ${\cal X}_A$ by changing $\tau_i$ to $\tau_i+1$, $\tau_n$ to $\tau_n+1$ and ${\vartheta}$ to ${\vartheta}+1$ for $k\in{\overline}{1,n-1}$.

<span>**Theorem 5.1**</span>. <span>*For $1\leq r_1<r_2\leq n-1$ and $r\in {\overline}{1,n-1}$, we have $${\partial}_{z_{r_2,r_1}}({\cal X}_A)=\sum_{s=1}^{r_1}\tau_sP_{[s,r_1]}{\cal X}_A[s,r_2],\eqno(5.8)$$ $${\partial}_{z_{n,r}}({\cal X}_A)=\frac{\tau_n}{{\vartheta}}\sum_{s=1}^r\tau_sP_{[s,r]}{\cal X}_A[s,n].
\eqno(5.9)$$*</span>

<span>*Proof*</span>. Note that $$\begin{aligned}
& & {\partial}_{z_{r_2,r_1}}({\cal X}_A)\\ &=& \sum_{{\beta}\in{\Gamma}_A}
\frac{\left[\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right]
(\tau_n)_{_{{\beta}_{{\underline}{n}}}}}{{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}{\beta}_{r_2,r_1}z^{{\beta}-{\epsilon}_{r_2,r_1}}\\
 &=&\sum_{{\beta}\in{\Gamma}_A}\frac{\left[(\tau_{r_1}-{\beta}_{{\underline}{r_1}})(\tau_{r_1}+1
 -{\beta}_{{\underline}{r_1}})_{{\beta}_{{\overline}{r_1}}}(\tau_{r_2}-1-{\beta}_{{\underline}{r_2}})_{{\beta}_{{\overline}{r_2}}}
 \prod_{s\neq r_1,r_2}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right](\tau_n)_{_{{\beta}_{{\underline}{n}}}}}
 {{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}z^{\beta}\\ &=&(\tau_{r_1}-\sum_{s=1}^{r_1-1}z_{r_1,s}{\partial}_{z_{r_1,s}})
 ({\cal X}_A[r_1,r_2]).\hspace{8.2cm}(5.10)\end{aligned}$$ In particular, $${\partial}_{z_{r,1}}({\cal X}_A)=\tau_1{\cal X}_A[1,r]\qquad{\mbox{for}}\;\;r\in{\overline}{1,n-1}.\eqno(5.11)$$ By (5.10), (5.11) and induction, we get (5.8). Moreover, $$\begin{aligned}
\qquad {\partial}_{z_{n,r}}({\cal X}_A)&=& \sum_{{\beta}\in{\Gamma}_A}\frac{\left[
\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right](\tau_n)_{_{{\beta}_{{\underline}{n}}}}}
{{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}{\beta}_{n,r}z^{{\beta}-{\epsilon}_{n,r}}\\ &=&
\sum_{{\beta}\in{\Gamma}_A}\frac{\left[(\tau_r-{\beta}_{{\underline}{r}})(\tau_r+1-{\beta}_{{\underline}{r}})_{{\beta}_{{\overline}{r}}}
\prod_{s\neq
r}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right]\tau_n(\tau_n+1)_{_{{\beta}_{{\underline}{n}}}}}
{{\beta}!{\vartheta}({\vartheta}+1)_{_{{\beta}_{{\underline}{n}}}}}z^{\beta}\\
&=&\frac{\tau_n}{{\vartheta}}(\tau_r-\sum_{s=1}^rz_{r,s}
{\partial}_{z_{r,s}})({\cal
X}_A[r,n]),\hspace{5.9cm}(5.12)\end{aligned}$$ by which we obtain (5.9).$\qquad\Box$

For any $z\in {\mathbb}{C}\setminus (-{\mathbb}{N})$, the gamma function $${\Gamma}(z)=\left[z e^{cz}\prod_{m=1}^{\infty}\left\{\left(1+\frac{z}{m}\right)e^{-z/m}\right\}
\right]^{-1},\eqno(5.13)$$ where $c$ is Euler's constant given by $$c=\lim_{m{\rightarrow}\infty}\left(\sum_{k=1}^m\frac{1}{k}-\ln m\right).\eqno(5.14)$$ When $\mbox{Re}\:z>0$, we have $${\Gamma}(z)=\int_0^{\infty}t^{z-1}e^{-t}dt.\eqno(5.15)$$ Now we have the following theorem of integral representation:

<span>**Theorem 5.2**</span>. <span>*Suppose $\mbox{\it Re}\:\tau_n>0$ and $\mbox{\it Re}\:({\vartheta}-\tau_n)>0$. We have*</span> $${\cal X}_A=\frac{{\Gamma}({\vartheta})}{{\Gamma}({\vartheta}-\tau_n){\Gamma}(\tau_n)}\int_0^1\left[\prod_{r=1}^{n-1}
(\sum_{s=r}^{n-1}P_{[r,s]}+tP_{[r,n]})^{-\tau_r}\right]t^{\tau_n-1}(1-t)^{{\vartheta}-\tau_n-1}dt.
\eqno(5.16)$$

<span>*Proof*</span>. For $\mu_1,\mu_2\in{\mathbb}{C}$ with $\mbox{Re}\:\mu_1,\mbox{Re}\:\mu_1>0$, we have the following Euler integral $$\int_0^1t^{\mu_1-1}(1-t)^{\mu_2-1}dt=\frac{{\Gamma}(\mu_1){\Gamma}(\mu_2)}{{\Gamma}(\mu_1+\mu_2)}.\eqno(5.17)$$ Moreover, $$(\mu)_m=\frac{{\Gamma}(\mu+m)}{{\Gamma}(\mu)}\qquad{\mbox{for}}\;\;m\in{\mathbb}{N},\;\mu\in{\mathbb}{C}.\eqno(5.18)$$ Recall $${\cal X}_A=\sum_{{\beta}\in{\Gamma}_A}\frac{\left[\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}
 \right](\tau_n)_{_{{\beta}_{{\underline}{n}}}}}{{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}z^{\beta}.\eqno(5.19)$$ We have $$\begin{aligned}
\qquad & &\frac{(\tau_n)_{_{{\beta}_{{\underline}{n}}}}}{({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}=
\frac{{\Gamma}({\vartheta})}{{\Gamma}({\vartheta}-\tau_n){\Gamma}(\tau_n)}\frac{{\Gamma}(\tau_n+{\beta}_{{\underline}{n}}){\Gamma}({\vartheta}-\tau_n)}{{\Gamma}({\vartheta}+{\beta}_{{\underline}{n}})}\\
&=&\frac{{\Gamma}({\vartheta})}{{\Gamma}({\vartheta}-\tau_n){\Gamma}(\tau_n)}\int_0^1t^{\tau_n+{\beta}_{{\underline}{n}}-1}
(1-t)^{{\vartheta}-\tau_n-1}dt\hspace{5.5cm}(5.20)\end{aligned}$$ by (5.17) and (5.19).

Denote $${\Gamma}_{(s)}=\sum_{1\leq r_1<r_2\leq s}{\mathbb}{N}{\epsilon}_{r_2,r_1},\qquad{\beta}_{{\overline}{r}}^s=\sum_{p=r+1}^s
{\beta}_{p,r}\qquad{\mbox{for}}\;\;{\beta}\in{\Gamma}_A,\;1\leq r<s\leq
n-1.\eqno(5.21)$$ Note that $$\begin{aligned}
& &\sum_{{\beta}\in{\Gamma}_A}\frac{\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}}
{{\beta}!}z^{\beta}\\&=&\sum_{{\beta}\in{\Gamma}_{(n-1)}}\frac{\prod_{s=1}^{n-2}(\tau_s-{\beta}_{{\underline}{s}})
_{_{{\beta}^{n-1}_{{\overline}{s}}}}}{{\beta}!}z^{\beta}\sum_{{\beta}_{n,1},...,{\beta}_{n,n-1}=0}^n\frac{(\tau_{n-1}
-{\beta}_{{\underline}{n-1}})_{{\beta}_{n,n-1}}z_{n,n-1}^{{\beta}_{n,n-1}}}{{\beta}_{n,n-1}!}\\
& &\times\prod_{r=1}^{n-2}
\frac{(\tau_r-{\beta}_{{\underline}{r}}+{\beta}^{n-1}_{{\overline}{r}})_{{\beta}_{n,r}}z_{n,r}^{{\beta}_{n,r}}}{{\beta}_{n,r}!}
\hspace{12cm}\end{aligned}$$ $$\begin{aligned}
 &=&
\sum_{{\beta}\in{\Gamma}_{(n-1)}}\frac{\prod_{s=1}^{n-2}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}^{n-1}_{{\overline}{s}}}}}
{{\beta}!}z^{\beta}(1-z_{n,n-1})^{{\beta}_{{\underline}{n-1}}-\tau_{n-1}}\prod_{r=1}^{n-2}(1-z_{n,r})^{{\beta}_{{\underline}{r}}
-{\beta}_{{\overline}{r}}^{n-1}-\tau_r}\\
&=& \left[\prod_{s=1}^{n-1}(1-z_{n,s})^{-\tau_s}\right]
\sum_{{\beta}\in{\Gamma}_{(n-1)}}\frac{\prod_{s=1}^{n-2}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}^{n-1}_{{\overline}{s}}}}}
{{\beta}!}\\
& &\times\prod_{1\leq r_1<r_2\leq
n-1}\left[\frac{(1-z_{n.r_2})z_{r_2,r_1}}{1-z_{n,r_1}}\right]^{{\beta}_{r_2,r_1}}.
\hspace{7.3cm}(5.22)\end{aligned}$$ Observe that $$1-\frac{(1-z_{n.n-1})z_{n-1,r}}{1-z_{n,r}}=\frac{1-z_{n-1,r}-z_{n,r}+z_{n.n-1}z_{n-1,r}}{1
-z_{n,r}}\qquad{\mbox{for}}\;\;r\in{\overline}{1,n-1}.\eqno(5.23)$$ By (5.22) and (5.23), we have $$\begin{aligned}
& &\sum_{{\beta}\in{\Gamma}_A}\frac{\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}}
{{\beta}!}z^{\beta}=(1-z_{n,n-1})^{-\tau_{n-1}}(1-z_{n-1,n-2}+P_{[n-2,n]})^{-\tau_{n-2}}\\
& &\times
\left[\prod_{s=1}^{n-2}(1-z_{n-1,s}-z_{n,s}+z_{n.n-1}z_{n-1,s})^{-\tau_s}\right]
\sum_{{\beta}\in{\Gamma}_{(n-2)}}\frac{\prod_{s=1}^{n-3}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}^{n-2}_{{\overline}{s}}}}}
{{\beta}!}\\ & &\times \prod_{1\leq r_1<r_2\leq
n-2}\left[\frac{(1-z_{n-1,r_2}-z_{n,r_2}
+z_{n.n-1}z_{n-1,r_2})z_{r_2,r_1}}{1-z_{n-1,r_1}-z_{n,r_1}+z_{n.n-1}z_{n-1,r_1}}
\right]^{{\beta}_{r_2,r_1}}.\hspace{3.3cm}(5.24)\end{aligned}$$ Now $$\begin{aligned}
 & &1-\frac{(1-z_{n-1,n-2}-z_{n,r-2}+z_{n.n-1}z_{n-1,n-2})z_{n-2,r}}
{1-z_{n-1,r}-z_{n,r}+z_{n.n-1}z_{n-1,r}}\\
&=&\frac{1}{1-z_{n-1,r}-z_{n,r}+z_{n.n-1}z_{n-1,r}}
[1-z_{n-2,r}-z_{n-1,r}+z_{n-1,n-2}z_{n-2,r}\\ &
&-z_{n,r}+z_{n.n-1}z_{n-1,r}+z_{n,r-2}z_{n-2,r}
-z_{n.n-1}z_{n-1,n-2}z_{n-2,r}].\hspace{3.7cm}(5.25)\end{aligned}$$ By induction, we can prove that $$\sum_{{\beta}\in{\Gamma}_A}\frac{\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}}{{\beta}!}z^{\beta}=\prod_{r=1}^{n-1}(\sum_{s=r}^nP_{[r,s]})^{-\tau_r}.\eqno(5.26)$$ Thus $$\sum_{{\beta}\in{\Gamma}_A}\frac{\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}}{{\beta}!}
z^{\beta}t^{{\beta}_{{\underline}{n}}}=\prod_{r=1}^{n-1}(\sum_{s=r}^{n-1}P_{[r,s]}+tP_{[r,n]})^{-\tau_r}.
\eqno(5.27)$$ Hence we obtain (5.16) by (5.20) and (5.27).$\qquad\Box$

<span>**Remark 5.3**</span>. According to \[M\], $$\left(\begin{array}{ccccc}1&0&0&\cdots&0\\ z_{2,1}&
1&0&\cdots&0\\ z_{3,1}&z_{3,2}&1&\ddots&\vdots\\
\vdots&\vdots&\ddots&\ddots&0\\ z_{n,1}&z_{n,2}&\cdots&z_{n,n-1}&
1\end{array}\right)^{-1}=\left(\begin{array}{ccccc}1&0&0&\cdots&0\\
P_{1,2}&
1&0&\cdots&0\\ P_{1,3}&P_{2,3}&1&\ddots&\vdots\\
\vdots&\vdots&\ddots&\ddots&0\\ P_{1,n}&P_{2,n}&\cdots&P_{n-1,n}&
1\end{array}\right).\eqno(5.28)$$

Denote $${\cal D}_{{\underline}{i}}^A=\sum_{r=1}^{i-1}z_{i,r}{\partial}_{z_{i,r}},\qquad {\cal D}_{{\overline}{i}}^A=
\sum_{s=i+1}^nz_{s,i}{\partial}_{z_{s,i}}\qquad{\mbox{for}}\;\;i\in{\overline}{1,n}.\eqno(5.29)$$

<span>**Theorem 5.4**</span>. <span>*We have: $$(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A+{\cal D}_{{\overline}{r_2}}^A){\partial}_{z_{r_2,r_1}}({\cal X}_A)
=(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A)(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A+{\cal D}_{{\overline}{r_1}}^A) ({\cal X}_A)\eqno(5.30)$$ for $1\leq r_1<r_2\leq n-1$ and $$({\vartheta}+{\cal D}_{{\underline}{n}}^A){\partial}_{z_{n,r}}({\cal X}_A)=(\tau_n+{\cal D}_{{\underline}{n}}^A)
(\tau_r-{\cal D}_{{\underline}{r}}^A+{\cal D}_{{\overline}{r}}^A)({\cal
X}_A)\eqno(5.31)$$ for $r\in{\overline}{1,n-1}$.*</span>

<span>*Proof*</span>. Let $$a_{\beta}=\frac{\left[\prod_{s=1}^{n-1}(\tau_s-{\beta}_{{\underline}{s}})_{_{{\beta}_{{\overline}{s}}}}\right]
(\tau_n)_{_{{\beta}_{{\underline}{n}}}}}{{\beta}!({\vartheta})_{_{{\beta}_{{\underline}{n}}}}}\qquad{\mbox{for}}\;\;{\beta}\in{\Gamma}_A.\eqno(5.32)$$ Then $$\frac{a_{{\beta}+{\epsilon}_{r_2,r_1}}}{a_{\beta}}=\frac{(\tau_{r_2}-{\beta}_{{\underline}{r_2}}-1)(\tau_{r_1}-
{\beta}_{{\underline}{r_1}}+{\beta}_{{\overline}{r_1}})}{(\tau_{r_2}-1-{\beta}_{{\underline}{r_2}}+{\beta}_{{\overline}{r_2}})({\beta}_{r_2,r_1}+1)},
\eqno(5.33)$$ equivalently, $$(\tau_{r_2}-1-{\beta}_{{\underline}{r_2}}+{\beta}_{{\overline}{r_2}})({\beta}_{r_2,r_1}+1)a_{{\beta}+{\epsilon}_{r_2,r_1}}=(\tau_{r_2}
-{\beta}_{{\underline}{r_2}}-1)(\tau_{r_1}-{\beta}_{{\underline}{r_1}}+{\beta}_{{\overline}{r_1}})a_{\beta}.\eqno(2.34)$$ Thus $$\sum_{{\beta}\in{\Gamma}_A}(\tau_{r_2}-1-{\beta}_{{\underline}{r_2}}+{\beta}_{{\overline}{r_2}})({\beta}_{r_2,r_1}+1)
a_{{\beta}+{\epsilon}_{r_2,r_1}}z^{\beta}=\sum_{{\beta}\in{\Gamma}_A}(\tau_{r_2}-{\beta}_{{\underline}{r_2}}-1)(\tau_{r_1}
-{\beta}_{{\underline}{r_1}}+{\beta}_{{\overline}{r_1}})a_{\beta}z^{\beta},\eqno(2.35)$$ which is equivalent to (5.30). Similarly, (5.31) follows from $$\frac{a_{{\beta}+{\epsilon}_{n,r}}}{a_{\beta}}=\frac{(\tau_n-{\beta}_{{\underline}{n}})(\tau_r-{\beta}_{{\underline}{r}}+{\beta}_{{\overline}{r}})}
{({\vartheta}+{\beta}_{{\underline}{n}})({\beta}_{n,r}+1)}.\qquad\Box\eqno(5.36)$$

Next we study our hypergeometric functions of type C. For convenience, we simply denote $${\cal X}_C={\cal X}_C(\tau_1,...,\tau_n;{\vartheta})\{z_{r_2,r_1},z_{n+s_2,s_1}\}\eqno(5.37)$$ (cf. (4.114)). First we have $$\begin{aligned}
& &{\partial}_{z_{r_2,r_1}}({\cal X}_C)\\ &=&\sum_{{\alpha}\in{\Gamma}_C}\frac{\prod_{r=1}^n
(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}{{\alpha}!({\vartheta})_{{\alpha}^c}}{\alpha}_{r_2,r_1}z^{{\alpha}-{\epsilon}_{r_2,r_1}}\hspace{9cm}\end{aligned}$$ $$\begin{aligned}
 &=& \sum_{{\alpha}\in{\Gamma}_C}\frac{\prod_{r\neq
r_1,r_2}(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}{{\alpha}!({\vartheta})_{{\alpha}^c}}(\tau_{r_1}
-{\alpha}_{{\underline}{r_1}})(\tau_{r_1}+1-{\alpha}_{{\underline}{r_1}})_{_{{\alpha}_{{\overline}{r_1}}+{\alpha}^c_{r_1}}}\\
&
&\times(\tau_{r_2}-1-{\alpha}_{{\underline}{r_2}})_{_{{\alpha}_{{\overline}{r_2}}+{\alpha}^c_{r_2}}}z^{\alpha}.
\hspace{9.3cm}(5.38)\end{aligned}$$ or $1\leq r_1<r_2\leq n$. Moreover, $$\begin{aligned}
& &{\partial}_{z_{n+r_2,r_1}}({\cal X}_C)\\ &=&\sum_{{\alpha}\in{\Gamma}_C}
\frac{\prod_{r=1}^{n-1}(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}{{\alpha}!({\vartheta})_{{\alpha}^c}}
{\alpha}_{n+r_2,r_1}z^{{\alpha}-{\epsilon}_{n+r_2,r_1}}\\ &=&
\frac{1}{{\vartheta}}\sum_{{\alpha}\in{\Gamma}_C} \frac{\prod_{r\neq
r_1,r_2}(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}
{{\alpha}!({\vartheta}+1)_{{\alpha}^c}}(\tau_{r_1}-{\alpha}_{{\underline}{r_1}})(\tau_{r_2}-{\alpha}_{{\underline}{r_2}})(\tau_{r_1}
+1-{\alpha}_{{\underline}{r_1}})_{_{{\alpha}_{{\overline}{r_1}}+{\alpha}^c_{r_1}}}\\ & &\times
(\tau_{r_2}+1-
{\alpha}_{{\underline}{r_2}})_{_{{\alpha}_{{\overline}{r_2}}+{\alpha}^c_{r_2}}}z^{{\alpha}}\hspace{9.3cm}(5.39)\end{aligned}$$ for $1\leq r_1<r_2\leq n$ and $$\begin{aligned}
{\partial}_{z_{n+s,s}}({\cal X}_C)&=&\sum_{{\alpha}\in{\Gamma}_C}\frac{\prod_{r=1}^n(\tau_r
-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}{{\alpha}!({\vartheta})_{{\alpha}^c}}{\alpha}_{n+s,s}z^{{\alpha}-{\epsilon}_{n+s,s}}\\
&=&\frac{1}{{\vartheta}}\sum_{{\alpha}\in{\Gamma}_C}\frac{\prod_{r\neq s}(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}
+{\alpha}^c_r}}}{{\alpha}!({\vartheta}+1)_{{\alpha}^c}}(\tau_{s}-{\alpha}_{{\underline}{s}})(\tau_{s}+1-{\alpha}_{{\underline}{s}})\\
& &\times
(\tau_s+2-{\alpha}_{{\underline}{s}})_{_{{\alpha}_{{\overline}{s}}+{\alpha}^c_{s}}}z^{{\alpha}}\hspace{8cm}(5.40)\end{aligned}$$ for $s\in{\overline}{1,n}$.

Expressions (5.38)-(5.40) motivate us to define $${\cal X}_C[i_1,i_2]={\cal X}_C(\tau_1,...,\tau_{i_1}+1,...,\tau_{i_2}-1,...\tau_n;{\vartheta})
\{z_{r_2,r_1},z_{n+s_2,s_1}\}\eqno(5.41)$$ obtained from ${\cal
X}_C$ by changing $\tau_{i_1}$ to $\tau_{i_1}+1$ and $\tau_{i_2}$ to $\tau_{i_2}-1$ for $1\leq i_1<i_2\leq n$, $${\cal X}_C[j_1,j_2;1]={\cal X}_C(\tau_1,...,\tau_{j_1}+1,...,\tau_{j_2}+1,...\tau_n;{\vartheta}+1)
\{z_{r_2,r_1},z_{n+s_2,s_1}\}\eqno(5.42)$$ obtained from ${\cal
X}_C$ by changing $\tau_{j_1}$ to $\tau_{j_1}+1$, $\tau_{j_2}$ to $\tau_{j_2}+1$ and ${\vartheta}$ to ${\vartheta}+1$ for $1\leq j_1<j_2\leq n$ and $${\cal X}_C[k(2)]={\cal X}_C(\tau_1,...,\tau_k+2,...,\tau_n;{\vartheta}+1)\{z_{r_2,r_1},z_{n+s_2,s_1}\}
\eqno(5.43)$$ obtained from ${\cal X}_C$ by changing $\tau_k$ to $\tau_k+2$ and ${\vartheta}$ to ${\vartheta}+1$ for $k\in{\overline}{1,n}$. Now (5.38) can be written as $${\partial}_{z_{r_2,r_1}}({\cal X}_C)=(\tau_{r_1}-\sum_{s=1}^{r_1-1}z_{r_1,s}{\partial}_{z_{r_1,s}})
({\cal X}_C[i_1,i_2])\eqno(5.44)$$ and (5.39) becomes $${\partial}_{z_{n+r_2,r_1}}({\cal X}_C)=\frac{1}{{\vartheta}}(\tau_{r_1}-\sum_{s_1=1}^{r_1-1}z_{r_1,s_1}
{\partial}_{z_{r_1,s_1}})(\tau_{r_2}-\sum_{s_2=1}^{r_2-1}z_{r_2,s_2}{\partial}_{z_{r_2,s_2}})
({\cal X}_C[i_1,i_2;1])\eqno(5.45)$$ for $1\leq r_1<r_2\leq n.$ Moreover, (5.40) is equivalent to $${\partial}_{z_{n+s,s}}({\cal X}_C)=\frac{1}{{\vartheta}}(\tau_s-\sum_{r=1}^{s-1}z_{s,r}{\partial}_{z_{s,r}})
(\tau_s+1-\sum_{r=1}^{s-1}z_{s,r}{\partial}_{z_{s,r}})({\cal
X}_C[s(2)])\eqno(5.46)$$ for $s\in{\overline}{1,n}$. By (5.44)-(5.46) and induction, we obtain:

<span>**Theorem 5.5**</span>. <span>*The following equations hold for ${\cal
X}_C$: $${\partial}_{z_{r_2,r_1}}({\cal X}_C)=\sum_{s=1}^{r_1}\tau_sP_{[s,r_1]}{\cal X}_C[s,r_2],
\eqno(5.47)$$ $$\begin{aligned}
{\partial}_{z_{n+r_2,r_1}}({\cal X}_C)&=&\frac{1}{{\vartheta}}[\sum_{i=1}^{r_1}\tau_i^2
P_{[i,r_1]}P_{[i,r_2]}{\cal
X}_C[i(2)]+\sum_{s_1=1}^{r_1}\sum_{s_2=r_1+1}^{r_2}\tau_{s_1}
\tau_{s_2}P_{[s_1,r_1]}P_{[s_2,r_2]}{\cal X}_C[s_1,s_2;1]\\ &
&+\sum_{1\leq s_1<s_2\leq r_1}
\tau_{s_1}\tau_{s_2}(P_{[s_1,r_1]}P_{[s_2,r_2]}+P_{[s_2,r_1]}P_{[s_1,r_2]})
{\cal X}_C[s_1,s_2;1]]\hspace{1cm}(5.48)\end{aligned}$$ for $1\leq
r_1<r_2\leq n$ and $$\begin{aligned}
{\partial}_{z_{n+s,s}}({\cal X}_C)&=&\frac{1}{{\vartheta}}[\sum_{i=1}^s\tau_i^2P_{[i,s]}^2
{\cal X}_C[i(2)]+\tau_s{\cal X}_C[s(2)]+
\sum_{i=1}^{s-1}\tau_iP_{[i,s]}{\cal X}_C[i,s;1]\\ & &
+2\sum_{1\leq r_1<r_2\leq
s}\tau_{r_1}\tau_{r_2}P_{[r_1,s]}P_{[r_2,s]}{\cal X}_C[r_1,r_2;1]]
\hspace{4.5cm}(5.49)\end{aligned}$$ for $s\in{\overline}{1,n}$.*</span>

Up to this stage, we have not found a nice integral representation for ${\cal X}_C$. In fact, there is no simple integral formula of Euler type with an elementary integrand for Lauricella third multiple hypergeometric function (e.g., cf. \[Eh\]). It might also be the case for our hypergeometric function ${\cal X}_C$.

By (4.114), we set $$c_{\alpha}=\frac{\prod_{r=1}^n(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^c_r}}}{{\alpha}!({\vartheta})_{{\alpha}^c}}
z^{\alpha}\qquad{\mbox{for}}\;\;{\alpha}\in{\Gamma}_C.\eqno(5.50)$$ Note that $$\frac{c_{{\alpha}+{\epsilon}_{r_2,r_1}}}{c_{\alpha}}=\frac{(\tau_{r_1}-{\alpha}_{_{{\underline}{r_1}}}+{\alpha}_{_{{\overline}{r_1}}}
+{\alpha}^c_{r_1})(\tau_{r_2}-1-{\alpha}_{_{{\underline}{r_2}}})}{({\alpha}_{r_2,r_1}+1)(\tau_{r_2}-1-{\alpha}_{_{{\underline}{r_2}}}
+{\alpha}_{_{{\overline}{r_2}}}+{\alpha}^c_{r_2})},\eqno(5.51)$$ $$\frac{c_{{\alpha}+{\epsilon}_{n+r_2,r_1}}}{c_{\alpha}}=\frac{(\tau_{r_1}-{\alpha}_{_{{\underline}{r_1}}}+{\alpha}_{_{{\overline}{r_1}}}
+{\alpha}^c_{r_1})(\tau_{r_2}-{\alpha}_{_{{\underline}{r_2}}}+{\alpha}_{_{{\overline}{r_2}}}+{\alpha}^c_{r_2})}{({\alpha}_{n+r_2,r_1}+1)
({\vartheta}+{\alpha}^c)}\eqno(5.52)$$ for $1\leq r_1<r_2\leq n$ and $$\frac{c_{{\alpha}+{\epsilon}_{n+s,s}}}{c_{\alpha}}=\frac{(\tau_s-{\alpha}_{_{{\underline}{s}}}+{\alpha}_{_{{\overline}{s}}}+{\alpha}^c_{s})
(\tau_s+1-{\alpha}_{_{{\underline}{s}}}+{\alpha}_{_{{\overline}{s}}}+{\alpha}^c_{s})}{({\alpha}_{n+s,s}+1)({\vartheta}+{\alpha}^c)}\eqno(5.53)$$ for $s\in{\overline}{1,n}$. Let $${\cal D}^C_r=\sum_{i=1}^rz_{n+r,i}{\partial}_{z_{n+r,i}}+\sum_{s=r}^nz_{n+s,r}{\partial}_{z_{n+s,r}}
\qquad{\mbox{for}}\;\;r\in{\overline}{1,n}\eqno(5.54)$$ and $${\cal D}^C=\sum_{1\leq r_1\leq r_2\leq n}z_{n+r_2,r_1}{\partial}_{z_{n+r_2,r_1}}.\eqno(5.55)$$ By the proof of Theorem 5.4, we have:

<span>**Theorem 5.6**</span>. <span>*The function ${\cal X}_C$ satisfies: $$(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A+{\cal D}_{{\overline}{r_2}}^A+{\cal D}^C_{r_2}){\partial}_{z_{r_2,r_1}}
({\cal X}_C)=(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A)(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A +{\cal D}_{{\overline}{r_1}}^A+{\cal D}_{r_1}^C)({\cal
X}_C),\eqno(5.56)$$ $$({\vartheta}+{\cal D}^C){\partial}_{z_{n+r_2,r_1}}({\cal X}_C)=(\tau_{r_2}-{\cal D}_{{\underline}{r_2}}^A
+{\cal D}_{{\overline}{r_2}}^A+{\cal D}^C_{r_2})(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A+{\cal D}_{{\overline}{r_1}}^A +{\cal D}_{r_1}^C)({\cal
X}_C)\eqno(5.57)$$ for $1\leq r_1<r_2\leq n$ and $$({\vartheta}+{\cal D}^C){\partial}_{z_{n+s,s}}({\cal X}_C)=(\tau_s-{\cal D}_{{\underline}{s}}^A+{\cal D}_{{\overline}{s}}^A
+{\cal D}^C_s)(\tau_s+1-{\cal D}_{{\underline}{s}}^A+{\cal
D}_{{\overline}{s}}^A+{\cal D}_{s}^C)({\cal X}_C) \eqno(5.58)$$ for $s\in{\overline}{1,n}$.*</span>

Set $${\Gamma}_D=\sum_{1\leq r_1<r_2\leq n}({\mathbb}{N}{\epsilon}_{r_2,r_1}+{\mathbb}{N}{\epsilon}_{n+r_2,r_1})\subset {\Gamma}_C.
\eqno(5.59)$$ Moreover, we let $${\beta}_r^D=\sum_{i=1}^{r-1}{\beta}_{n+r,i}+\sum_{s=r+1}^n{\beta}_{n+s,r},\;\;{\beta}^D=\sum_{1\leq r_1<r_2
\leq n}{\beta}_{n+r_2,r_1}\qquad{\mbox{for}}\;\;{\beta}\in{\Gamma}_D\eqno(5.60)$$ and $${\beta}_r^B=\sum_{i=1}^r{\beta}_{n+r,i}+\sum_{s=r+1}^n{\beta}_{n+s,r},
\;\;{\beta}^B={\beta}^c=\sum_{1\leq r_1\leq r_2\leq
n}{\beta}_{n+r_2,r_1}\qquad{\mbox{for}}\;\;{\beta}\in{\Gamma}_C. \eqno(5.61)$$ We define the following hypergeometric functions: $${\cal X}_D(\tau_1,...,\tau_n;{\vartheta})\{z_{r_2,r_1},z_{n+r_2,r_1}\}=\sum_{{\alpha}\in{\Gamma}_D}
\frac{\prod_{r=1}^n(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^D_r}}}{{\alpha}!({\vartheta})_{{\alpha}^D}}z^{\alpha}.
\eqno(5.62)$$ $${\cal X}_B(\tau_1,...,\tau_n;{\vartheta})\{z_{r_2,r_1},z_{n+s_2,s_1}\}=\sum_{{\alpha}\in{\Gamma}_C}
\frac{\prod_{r=1}^n(\tau_r-{\alpha}_{{\underline}{r}})_{_{{\alpha}_{{\overline}{r}}+{\alpha}^B_r}}}{{\alpha}!({\vartheta})_{{\alpha}^B}}z^{\alpha}.
\eqno(5.63)$$ Furthermore, we define ${\cal X}_B,\;{\cal
X}_B[i_1,i_2],\;{\cal X}_B[j_1,j_2;1],\;{\cal X}_B[k(2)]$ and ${\cal X}_D,\;{\cal X}_D[i_1,i_2],\;{\cal X}_D[j_1,j_2;1],$ ${\cal
X}_D[k(2)]$ as those of type C in (5.37) and (5.41)-(5.43). We let $${\cal X}_B[k]={\cal X}_N(\tau_1,...,\tau_k+2,...,\tau_n;{\vartheta}+1)\{z_{r_2,r_1},z_{n+s_2,s_1}\}
\eqno(5.64)$$ obtained from ${\cal X}_B$ by changing $\tau_k$ to $\tau_k+1$ and ${\vartheta}$ to ${\vartheta}+1$ for $k\in{\overline}{1,n}$.

<span>**Theorem 5.7**</span>. <span>*The following equations hold for ${\cal
X}_C$: $${\partial}_{z_{r_2,r_1}}({\cal X}_B)=\sum_{s=1}^{r_1}\tau_sP_{[s,r_1]}{\cal X}_B[s,r_2],
\eqno(5.65)$$ $$\begin{aligned}
{\partial}_{z_{n+r_2,r_1}}({\cal X}_B)&=&\frac{1}{{\vartheta}}[\sum_{i=1}^{r_1}
\tau_i^2P_{[i,r_1]}P_{[i,r_2]}{\cal
X}_B[i(2)]+\sum_{s_1=1}^{r_1}\sum_{s_2=r_1+1}^{r_2}
\tau_{s_1}\tau_{s_2}P_{[s_1,r_1]}P_{[s_2,r_2]}{\cal
X}_B[s_1,s_2;1]\\ & &+\sum_{1\leq s_1<s_2 \leq
r_1}\tau_{s_1}\tau_{s_2}(P_{[s_1,r_1]}P_{[s_2,r_2]}+P_{[s_2,r_1]}P_{[s_1,r_2]})
{\cal X}_B[s_1,s_2;1]],\hspace{0.9cm}(5.66)\end{aligned}$$ $${\partial}_{z_{r_2,r_1}}({\cal X}_D)=\sum_{s=1}^{r_1}\tau_sP_{[s,r_1]}{\cal X}_D[s,r_2],\eqno(5.67)$$ $$\begin{aligned}
{\partial}_{z_{n+r_2,r_1}}({\cal X}_D)&=&\frac{1}{{\vartheta}}[\sum_{i=1}^{r_1}
\tau_i^2P_{[i,r_1]}P_{[i,r_2]}{\cal
X}_D[i(2)]+\sum_{s_1=1}^{r_1}\sum_{s_2=r_1+1}^{r_2}
\tau_{s_1}\tau_{s_2}P_{[s_1,r_1]}P_{[s_2,r_2]}{\cal
X}_D[s_1,s_2;1]\\ & &+ \sum_{1\leq s_1<s_2\leq
r_1}\tau_{s_1}\tau_{s_2}(P_{[s_1,r_1]}P_{[s_2,r_2]}+
P_{[s_2,r_1]}P_{[s_1,r_2]}){\cal
X}_D[s_1,s_2;1]],\hspace{0.9cm}(5.68)\end{aligned}$$ for $1\leq
r_1<r_2\leq n$ and $${\partial}_{z_{n+s,s}}({\cal X}_B)=\sum_{r=1}^s\tau_rP_{[s,r]}X_B[s]\eqno(5.69)$$ for $s\in{\overline}{1,n}$.*</span>

Let $${\cal D}^B_r=\sum_{i=1}^rz_{n+r,i}{\partial}_{z_{n+r,i}}+\sum_{s=r+1}^nz_{n+s,r}{\partial}_{z_{n+s,r}},
\eqno(5.70)$$ $${\cal D}^D_r=\sum_{i=1}^{r-1}z_{n+r,i}{\partial}_{z_{n+r,i}}+\sum_{s=r+1}^nz_{n+s,r}
{\partial}_{z_{n+s,r}}\eqno(5.71)$$ for $r\in{\overline}{1,n}$ and $${\cal D}^B={\cal D}^C,\;\;{\cal D}^D=\sum_{1\leq r_1< r_2\leq n}z_{n+r_2,r_1}
{\partial}_{z_{n+r_2,r_1}}.\eqno(5.72)$$ As Theorem 5.4, we have:

<span>**Theorem 5.7**</span>. <span>*The functions ${\cal X}_B$ and ${\cal X}_C$ satisfy: $$(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A+{\cal D}_{{\overline}{r_2}}^A+{\cal D}^B_{r_2}){\partial}_{z_{r_2,r_1}}
({\cal X}_B)=(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A)(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A+ {\cal D}_{{\overline}{r_1}}^A+{\cal D}_{r_1}^B)({\cal
X}_B),\eqno(5.73)$$ $$(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A+{\cal D}_{{\overline}{r_2}}^A+{\cal D}^D_{r_2}){\partial}_{z_{r_2,r_1}}
({\cal X}_D)=(\tau_{r_2}-1-{\cal D}_{{\underline}{r_2}}^A)(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A+ {\cal D}_{{\overline}{r_1}}^A+{\cal D}_{r_1}^D)({\cal
X}_D),\eqno(5.74)$$ $$({\vartheta}+{\cal D}^B){\partial}_{z_{n+r_2,r_1}}({\cal X}_B)=(\tau_{r_2}-{\cal D}_{{\underline}{r_2}}^A
+{\cal D}_{{\overline}{r_2}}^A+{\cal D}^B_{r_2})(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A+{\cal D}_{{\overline}{r_1}}^A +{\cal D}_{r_1}^B)({\cal
X}_B),\eqno(5.75)$$ $$({\vartheta}+{\cal D}^D){\partial}_{z_{n+r_2,r_1}}({\cal X}_D)=(\tau_{r_2}-{\cal D}_{{\underline}{r_2}}^A
+{\cal D}_{{\overline}{r_2}}^A+{\cal D}^D_{r_2})(\tau_{r_1}-{\cal
D}_{{\underline}{r_1}}^A+{\cal D}_{{\overline}{r_1}}^A +{\cal D}_{r_1}^D)({\cal
X}_D)\eqno(5.76)$$ for $1\leq r_1<r_2\leq n$ and $$({\vartheta}+{\cal D}^B){\partial}_{z_{n+s,s}}({\cal X}_B)=(\tau_s-{\cal D}_{{\underline}{s}}^A+{\cal D}_{{\overline}{s}}^A
+{\cal D}^B_s)({\cal X}_B)\eqno(5.77)$$ for $s\in{\overline}{1,n}$.*</span>

Weyl Functions
==============

In this section, we show that certain variations of the Weyl functions of the finite-dimensional simple Lie algebras of types A, B, C and D give rise to special solutions of the Olshanesky-Perelomov equation (1.13) with different constants $K_1$-$K_4$ and $\nu$. Part of these results are already known. Nevertheless, we gave a self-contained complete exposition for reader's convenience. Like Theorem 2.2, the results in this section and further study on the systems of partial differential equations in last section would yield more solutions of the Olshanesky-Perelomov equation (1.13).

Let $f_{i,j}(z)\mid i,j\in{\overline}{1,n}\}$ be a set of one-variable differentiable functions and let $d_i$ be a one-variable differential operator in $z_i$ for $i\in{\overline}{1,n}$. It is easy to verified the following lemma:

<span>**Lemma 6.1**</span>. <span>*We have the following equation on differentiation of determinants*</span>: $$\begin{aligned}
\qquad & &(\sum_{i=1}^nd_i)\left(\left|\begin{array}{cccc}f_{1,1}(z_1)&
f_{1,2}(z_2)&\cdots & f_{1,n}(z_n)\\ f_{2,1}(z_1)& f_{2,2}(z_2)&\cdots & f_{2,n}(z_n)\\
 \vdots&\vdots&\vdots&\vdots\\ f_{n,1}(z_1)& f_{n,2}(z_2)&\cdots &f_{n,n}(z_n)\end{array}
 \right|\right)\hspace{6cm}\end{aligned}$$ $$\begin{aligned}
\qquad &=&\sum_{i=1}^n\left|\begin{array}{cccc}f_{1,1}(z_1)& f_{1,2}(z_2)&
\cdots & f_{1,n}(z_n)\\ \vdots&\vdots&\vdots&\vdots\\ f_{i-1,1}(z_1)& f_{i-1,2}(z_2)&\cdots &
f_{i-1,n}(z_n)\\ d_1(f_{i,1}(z_1))& d_2(f_{i,2}(z_2))&\cdots & d_n(f_{i,n}(z_n))
\\ f_{i+1,1}(z_1)& f_{i+1,2}(z_2)&\cdots & f_{i+1,n}(z_n)\\ \vdots&\vdots&\vdots&\vdots\\
 f_{n,1}(z_1)& f_{n,2}(z_2)&\cdots & f_{n,n}(z_n)\end{array}\right|.\hspace{4.3cm}(6.1)
 \end{aligned}$$

Suppose that $R^+_X\subset {\mathbb}{R}^n$ is the set of positive roots of the finite-dimensional simple Lie algebra of type X (e.g. cf. Section 12.1 in \[Hj\]). We denote $$z^{\alpha}=\prod_{i=1}^nz_i^{{\alpha}_i}\qquad{\mbox{for}}\;\;{\alpha}\in{\mathbb}{R}^n.\eqno(6.2)$$ The Weyl function $${\cal W}_X=\prod_{{\alpha}\in R^+_X}(z^{{\alpha}/2}-z^{-{\alpha}/2}).\eqno(6.3)$$

First the Weyl function of $sl(n)$ is $${\cal W}_{A_{n-1}}=\prod_{1\leq i<j\leq n}(z_i^{1/2}z_j^{-1/2}-z_i^{-1/2}z_j^{1/2})=
(z_1z_2\cdots z_n)^{(1-n)/2}\prod_{1\leq i<j\leq n}(z_i-z_j),\eqno(6.4)$$ Denote the Vandermonde determinant $$W(z_1,z_2,...,z_n)=\left|\begin{array}{cccc}1&1&\cdots &1\\ z_1&z_2&\cdots &z_n\\ z_1^2&z_2^2
&\cdots &z_n^2 \\ \vdots&\vdots&\vdots&\vdots\\ z_1^{n-1}&z_2^{n-1}&\cdots &z_n^{n-1}
\end{array}\right|=\prod_{1\leq i<j\leq n}(z_i-z_j).\eqno(6.5)$$ Then $W(z_1,z_2,...,z_n)$ is the fundamental part of ${\cal W}_{A_{n-1}}$. By Lemma 6.1, we have $$\begin{aligned}
 (\sum_{i=1}^n(z_i{\partial}_{z_i})^2)(W(z_1,z_2,...,z_n))&=&(\sum_{i=1}^{n-1}i^2)
W(z_1,z_2,...,z_n)\\ &=&\frac{(n-1)n(2n-1)}{6}W(z_1,z_2,...,z_n).\hspace{2.9cm}(6.6)\end{aligned}$$ On the other hand, $$\begin{aligned}
 & &(\sum_{i=1}^n(z_i{\partial}_{z_i})^2)(W(z_1,z_2,...,z_n))\\ &=&(\sum_{r=1}^n
(z_r{\partial}_{z_r})^2)(\prod_{1\leq i<j\leq n}(z_i-z_j))\\
&=&\sum_{r=1}^nz_r\left[-\sum_{s=1}^{r-1} \left(\prod_{1\leq
i<j\leq n;\:(i,j)\neq (s,r)}(z_i-z_j)\right)+\sum_{s=r+1}^n
\left(\prod_{1\leq i<j\leq n;\:(i,j)\neq
(r,s)}(z_i-z_j)\right)\right]\\ & &+2\sum_{r=1}^n
z_r^2(\sum_{1\leq s_1<s_2<r}\left[\prod_{1\leq i<j\leq
n;\:(i,j)\neq (s_1,r),(s_2,r)}(z_i-z_j) \right]\\ & &+
\sum_{r<s_1<s_2\leq n}\left[\prod_{1\leq i<j\leq n;\:(i,j)\neq
(r,s_1),(r,s_2)}(z_i-z_j)\right]
\\ & &-\sum_{1\leq s_1<r<s_2\leq n}\left[\prod_{1\leq i<j\leq n;\:(i,j)\neq (s_1,r),(r,s_2)}
(z_i-z_j)\right])\\ &=&\sum_{1\leq s<r\leq n}(z_s-z_r)\prod_{1\leq i<j\leq n;\:(i,j)\neq (s,r)}
(z_i-z_j)\\ & &+2W(z_1,z_2,...,z_n)\sum_{r=1}^n\sum_{1\leq s_1<s_2\leq n;\:s_1,s_2\neq r}
\frac{z_r^2}{(z_{s_1}-z_r)(z_{s_2}-z_r)}\\ &=&\left(\frac{n(n-1)}{2}+2
\sum_{r=1}^n\sum_{1\leq s_1<s_2\leq n;\:s_1,s_2\neq r}\frac{z_r^2}{(z_{s_1}-z_r)(z_{s_2}-z_r)}
\right)W(z_1,z_2,...,z_n).\hspace{1cm}(6.7)\end{aligned}$$ Thus (6.6) and (6.7) yield $$\begin{aligned}
\qquad\qquad& &\sum_{r=1}^n\sum_{1\leq s_1<s_2\leq n;\:s_1,s_2\neq r}
\frac{z_r^2}{(z_{s_1}-z_r)(z_{s_2}-z_r)}\\
&=&\frac{1}{2}\left[\frac{(n-1)n(2n-1)}{6}
-\frac{n(n-1)}{2}\right]\\
&=&\frac{(n-1)n(n-2)}{6}=\left(\!\!\begin{array}{c}n\\
3\end{array} \!\!\right).\hspace{7.9cm}(6.8)\end{aligned}$$

Let $$\phi^A_{\mu_1,\mu_2}=(z_1z_2\cdots z_n)^{\mu_1}W^{\mu_2}(z_1,z_2,...,z_n)\qquad{\mbox{for}}\;\;
\mu_1,\mu_2\in{\mathbb}{C}.\eqno(6.9)$$ Then $$z_r{\partial}_{z_r}(\phi^A_{\mu_1,\mu_2})=\left(\mu_1-\mu_2\sum_{s=1}^{r-1}\frac{z_r}{z_s-z_r}
+\mu_2\sum_{s=r+1}^n\frac{z_r}{z_r-z_s}\right)\phi^A_{\mu_1,\mu_2}\eqno(6.10)$$ for $r\in{\overline}{1,n}$. Thus $$\begin{aligned}
& &\sum_{r=1}^n(z_r{\partial}_{z_r})^2(\phi^A_{\mu_1,\mu_2})\\ &=&\sum_{r=1}^n
[\mu_1^2-2\mu_1\mu_2\sum_{s=1}^{r-1}\frac{z_r}{z_s-z_r}+2\mu_1\mu_2\sum_{s=r+1}^n\frac{z_r}
{z_r-z_s}-\mu_2\sum_{r\neq s\in{\overline}{1,n}}\frac{z_sz_r}{(z_s-z_r)^2}\\
&&+\mu_2^2\sum_{r\neq
s\in{\overline}{1,n}}\frac{z_r^2}{(z_s-z_r)^2}+2\mu_2^2 \sum_{1\leq
s_1<s_2\leq n;\:s_1,s_2\neq
r}\frac{z_r^2}{(z_{s_1}-z_r)(z_{s_2}-z_r)}] \phi^A_{\mu_1,\mu_2}\\
&=&[n\mu_1^2+n(n-1)\mu_1\mu_2+\mu_2^2
\left(\!\!\begin{array}{c}n\\
3\end{array}\!\!\right)-2\mu_2\sum_{1\leq r<s\leq n}
\frac{z_sz_r}{(z_s-z_r)^2}\\ & &+2\mu_2^2\sum_{1\leq r<s\leq
n}\frac{z_r^2+z_s^2}
{(z_s-z_r)^2}]\phi^A_{\mu_1,\mu_2}\\&=&[n\mu_1^2+n(n-1)\mu_1\mu_2+2\mu_2^2
\left(\!\!\begin{array}{c}n\\
3\end{array}\!\!\right)-2\mu_2\sum_{1\leq r<s\leq n}
\frac{z_sz_r}{(z_s-z_r)^2}\\ & &+\mu_2^2\sum_{1\leq r<s\leq
n}\frac{z_r^2+z_s^2-2z_rz_s+2z_rz_s}{(z_s-z_r)^2}]\phi^A_{\mu_1,\mu_2}\\
&=&[n\mu_1^2+n(n-1)(\mu_1+\mu_2/2)\mu_2+2\mu_2^2\left(\!\!\begin{array}{c}n\\
3\end{array}\!\!\right)\\ & &+2\mu_2(\mu_2-1)\sum_{1\leq r<s\leq
n}\frac{z_sz_r}{(z_s-z_r)^2}]\phi^A_{\mu_1,\mu_2}\hspace{7.2cm}(6.11)\end{aligned}$$ by (6.8) and (6.10). Therefore, we have:

<span>**Theorem 6.2**</span>. <span>*The function $\phi^A_{\mu_1,\mu_2}$ satisfies: $$\begin{aligned}
\qquad& &\sum_{r=1}^n(z_r{\partial}_{z_r})^2(\phi^A_{\mu_1,\mu_2})+2\mu_2(1-\mu_2)
\left(\sum_{1\leq i<j\leq n}\frac{z_iz_j}{(z_i-z_j)^2}\right)\phi^A_{\mu_1,\mu_2}\\
&=&\left[n\mu_1^2+n(n-1)(\mu_1+\mu_2/2)\mu_2+2\left(\!\!\begin{array}{c}n\\ 3\end{array}
\!\!\right)\mu_2^2\right]\phi^A_{\mu_1,\mu_2},\hspace{4cm}(6.12)\end{aligned}$$ which is a Calogero-Sutherland equation.*</span>

We remark that above result was known when $\mu_1=\mu_2$ or $\mu_1=0$. Moreover, ${\cal W}_{A_n}=\phi^A_{(1-n)/2,1}$.

According to simplicity, we secondly consider the Weyl function of $so(2n)$. By the root structure, $$\begin{aligned}
\qquad{\cal W}_{D_n}&=&\prod_{1\leq i<j\leq n}(z_i^{1/2}z_j^{-1/2}-z_i^{-1/2}
z_j^{1/2})(z_i^{1/2}z_j^{1/2}-z_i^{-1/2}z_j^{-1/2})\\ &=&
\prod_{1\leq i<j\leq n}(z_i+z_i^{-1}-(z_j+z_j^{-1}))\\
&=&W(z_1+z_1^{-1},z_2+z_2^{-1},...,z_n
+z_n^{-1})\hspace{5.8cm}(6.13)\end{aligned}$$(cf. (6.5)). Note that $$\begin{aligned}
(z_i{\partial}_{z_i})^2[(z_i+z_i^{-1})^k]&=&k(z_i+z_i^{-1})^k+k(k-1)(z_i-z_i^{-1})^2
(z_i+z_i^{-1})^{k-2}\\ &=& k(z_i+z_i^{-1})^k+k(k-1)(z_i^2-2+z_i^{-2})(z_i+z_i^{-1})^{k-2}\\
 &=& k^2(z_i+z_i^{-1})^k-4k(k-1)(z_i+z_i^{-1})^{k-2}.\hspace{3.4cm}(6.14)\end{aligned}$$ By Lemma 6.1 and a similar calculation as (6.5), $$(\sum_{i=1}^n(z_i{\partial}_{z_i})^2)({\cal W}_{D_n})=\frac{(n-1)n(2n-1)}{6}{\cal W}_{D_n}.
\eqno(6.15)$$ On the other hand, by (6.7) and (6.14), we have $$\begin{aligned}
 & &\sum_{r=1}^n\sum_{1\leq s_1<s_2\leq n;\:s_1,s_2\neq r}
\frac{(z_r-z_r^{-1})^2}{(z_{s_1}+z_{s_1}^{-1}-z_r-z_r^{-1})(z_{s_2}+z_{s_2}^{-1}-z_r-z_r^{-1})}
\\ &=&\frac{(n-1)n(n-2)}{6}=\left(\!\!\begin{array}{c}n\\ 3\end{array}\!\!\right).
\hspace{9.4cm}(6.16)\end{aligned}$$ Moreover, $$\begin{aligned}
 & &\frac{(z_r-z_r^{-1})^2+(z_s-z_s^{-1})^2}{(z_s+z_s^{-1}-z_r-z_r^{-1})^2}\\
 &=& \frac{z_r^2+z_r^{-2}+z_s^2+z_s^{-2}-4}{(z_rz_s)^{-2}(z_s-z_r)^2(z_rz_s-1)^2}\\ &=&
 \frac{z_r^4z_s^2+z_s^2+z_r^2z_s^4+z_s^2-4z_r^2z_s^2}{(z_s-z_r)^2(z_rz_s-1)^2}\\ &=&
\frac{1}{(z_s-z_r)^2(z_rz_s-1)^2}
 [z_r^4z_s^2+z_s^2+z_r^2z_s^4+z_s^2-2z_rz_s(z_r^2+z_s^2+(z_rz_s)^2-2z_rz_s+1)\\ & &
 +2z_rz_s(z_r^2+z_s^2+(z_rz_s)^2-4z_rz_s+1)]\\ &=&\frac{(z_s-z_r)^2(z_rz_s-1)^2
 +2z_rz_s[(z_r-z_s)^2+(z_rz_s-1)^2]}{(z_s-z_r)^2(z_rz_s-1)^2}\\ &=&1 +2\frac{z_rz_s}
 {(z_r-z_s)^2}+2\frac{z_rz_s}{(z_rz_s-1)^2}\hspace{8cm}(6.17)\end{aligned}$$ and $$\begin{aligned}
\qquad& &z_r{\partial}_{z_r}\left(\frac{z_r-z_r^{-1}}{z_r+z_r^{-1}-z_s-z_s^{-1}}
\right)\\ &=&\frac{(z_r+z_r^{-1})(z_r+z_r^{-1}-z_s-z_s^{-1})-(z_r-z_r^{-1})^2}{(z_r+z_r^{-1}
-z_s-z_s^{-1})^2}\\ &=& \frac{(z_r+z_r^{-1})^2-(z_r+z_r^{-1})(z_s+z_s^{-1})-(z_r-z_r^{-1})^2}
{(z_r+z_r^{-1}-z_s-z_s^{-1})^2}\\ &=&
 \frac{4-z_rz_s-z_rz_s^{-1}-z_r^{-1}z_s-z_r^{-1}z_s^{-1}}{(z_rz_s)^{-2}(z_s-z_r)^2(z_rz_s-1)^2}
 \\ &=& -z_rz_s\frac{(z_rz_s)^2+z_r^2+z_s^2+1-4z_rz_s}{(z_s-z_r)^2(z_rz_s-1)^2}\\ &=&
 -\frac{z_rz_s}{(z_s-z_r)^2}-\frac{z_rz_s}{(z_rz_s-1)^2}\hspace{8.4cm}(6.18)\end{aligned}$$ for $r,s\in{\overline}{1,n}$ such that $r\neq s$.

Set $$\phi_\mu^D=({\cal W}_{D_n})^\mu=\prod_{1\leq i<j\leq n}(z_i+z_i^{-1}-z_j-z_j^{-1})^{\mu}
\qquad{\mbox{for}}\;\;\mu\in{\mathbb}{C}.\eqno(6.19)$$ Then $$\begin{aligned}
 & &(\sum_{r=1}^n(z_r{\partial}_{z_r})^2)(\phi_\mu^D)\\ &=&\sum_{r=1}^n
[\sum_{r\neq s\in{\overline}{1,n}}\left[\mu
z_r{\partial}_{z_r}\left(\frac{z_r-z_r^{-1}}
{z_r+z_r^{-1}-z_s-z_s^{-1}}\right)+\mu^2\frac{(z_r-z_r^{-1})^2}{(z_s+z_s^{-1}-z_r-z_r^{-1})^2}
\right]\\ & &+2\mu^2\sum_{1\leq s_1<s_2\leq n;\:s_1,s_2\neq
r}\frac{(z_r-z_r^{-1})^2}{(z_{s_1}
+z_{s_1}^{-1}-z_r-z_r^{-1})(z_{s_2}+z_{s_2}^{-1}-z_r-z_r^{-1})}]\phi_\mu^D\\
&=& [\sum_{1\leq r<s\leq
n}\left[-2\mu\left(\frac{z_rz_s}{(z_s-z_r)^2}+\frac{z_rz_s}{(z_rz_s-1)^2}
\right)+\mu^2\frac{(z_r-z_r^{-1})^2+(z_s-z_s^{-1})^2}{(z_s+z_s^{-1}-z_r-z_r^{-1})^2}\right]
\phi_\mu^D\\&&+2\mu^2\sum_{r=1}^n\;\sum_{1\leq s_1<s_2\leq
n;\:s_1,s_2\neq r}
\frac{(z_r-z_r^{-1})^2}{(z_{s_1}+z_{s_1}^{-1}-z_r-z_r^{-1})(z_{s_2}+z_{s_2}^{-1}-z_r-z_r^{-1})}]
\\ &=& \left[2\mu(\mu-1)\sum_{1\leq r<s\leq n}\left(\frac{z_rz_s}{(z_s-z_r)^2}+\frac{z_rz_s}
{(z_rz_s-1)^2}\right)+\mu^2\left(\frac{n(n-1)}{2}+2\left(\!\!\begin{array}{c}n\\
3\end{array} \!\!\right)\right)\right]\phi_\mu^D\\
&=&\left[2\mu(\mu-1)\sum_{1\leq r<s\leq
n}\left(\frac{z_rz_s}{(z_s-z_r)^2}+\frac{z_rz_s}{(z_rz_s-1)^2}\right)
+\frac{n(n-1)(2n-1)}{6}\mu^2\right]\phi_\mu^D.\hspace{0.5cm}(6.20)\end{aligned}$$ Thus we obtain:

<span>**Theorem 6.3**</span>. <span>*The function $\phi_\mu^D$ the equation: $$\begin{aligned}
& &\sum_{r=1}^n(z_r{\partial}_{z_r})^2(\phi^D_\mu)+2\mu(1-\mu)
\left(\sum_{1\leq i<j\leq n}\frac{z_iz_j}{(z_i-z_j)^2}+\sum_{1\leq i<j\leq n}\frac{z_iz_j}
{(z_iz_j-1)^2}\right)\phi^D_\mu\\ &=&\frac{n(n-1)(2n-1)\mu^2}{6}\phi_\mu^D,
\hspace{9.7cm}(6.21)\end{aligned}$$ which is an Olshanesky-Perelomov equation*</span>.

We remark that the above result with $n=4$ was given in \[NFP\].

The Weyl function of $sp(2n)$ is $${\cal W}_{C_n}=\left[\prod_{r=1}^n(z_r-z_r^{-1})\right]\prod_{1\leq i<j\leq n}
(z_i+z_i^{-1}-(z_j+z_j^{-1})),\eqno(6.22)$$ In terms of determinant, $${\cal W}_{C_n}=\left|\begin{array}{ccc}z_1-z_1^{-1}&\cdots& z_n-z_n^{-1}\\ (z_1-z_1^{-1})
(z_1+z_1^{-1})&\cdots& (z_n-z_n^{-1})(z_n+z_n^{-1})\\ (z_1-z_1^{-1})(z_1+z_1^{-1})^2&
\cdots& (z_n-z_n^{-1})(z_n+z_n^{-1})^2\\ \vdots&\vdots&\vdots\\ (z_1-z_1^{-1})
(z_1+z_1^{-1})^{n-1}&\cdots& (z_n-z_n^{-1})(z_n+z_n^{-1})^{n-1}\end{array}\right|.
\eqno(6.23)$$ Moreover, $$\begin{aligned}
 (z_r{\partial}{z_r})^2[(z_r-z_r^{-1})(z_r+z_r^{-1})^k]&=&(k+1)^2(z_r-z_r^{-1})
(z_r+z_r^{-1})^k\\ &
&+4k(k-1)(z_r-z_r^{-1})(z_r+z_r^{-1})^{k-2}\hspace{2.3cm}(6.24)\end{aligned}$$ by (6.14). Thus $$(\sum_{i=1}^n(z_i{\partial}_{z_i})^2)({\cal W}_{C_n})=\frac{n(n+1)(2n+1)}{6}{\cal W}_{C_n}
\eqno(6.25)$$ by Lemma 6.1 and (6.24).

Set $$\psi^C_\mu=\prod_{r=1}^n(z_r-z_r^{-1})^{\mu}\qquad{\mbox{for}}\;\;\mu\in{\mathbb}{C}.\eqno(6.26)$$ Then $$\begin{aligned}
\sum_{r=1}^n(z_r{\partial}{z_r})^2(\psi^C_\mu)&=&\mu\psi^C_\mu\sum_{r=1}^n
\left[z_r{\partial}_{z_r}\left(\frac{z_r+z_r^{-1}}{z_r-z_r^{-1}}\right)+\mu\left(\frac{z_r+z_r^{-1}}
{z_r-z_r^{-1}}\right)^2\right]\\ &=&
\mu\psi^C_\mu\sum_{r=1}^n\left[\frac{(z_r-z_r^{-1})^2-(z_r+z_r^{-1})^2}{(z_r-z_r^{-1})^2}
+\mu\frac{(z_r+z_r^{-1})^2}{(z_r-z_r^{-1})^2}\right]\\
&=&\mu\sum_{r=1}^n\left[\mu+4(\mu-1)
\frac{1}{(z_r-z_r^{-1})^2}\right]\psi^C_\mu\\
&=&\left[n\mu^2+4\mu(\mu-1)\sum_{r=1}^n
\frac{z_r^2}{(z_r^2-1)^2}\right]\psi^C_\mu.\hspace{4.5cm}(6.27)\end{aligned}$$ Moreover, $$\begin{aligned}
\qquad& &\sum_{r=1}^n(z_r{\partial}{z_r})(\psi^C_{\mu_1})(z_r{\partial}{z_r})(\phi_{\mu_2}
^D)\\
&=&\mu_1\mu_2\sum_{r=1}^n\frac{z_r+z_r^{-1}}{z_r-z_r^{-1}}\psi^C_{\mu_1}
\sum_{r\neq
s\in{\overline}{1,n}}\frac{z_r-z_r^{-1}}{z_r+z_r^{-1}-z_s-z_s^{-1}}\phi_{\mu_2}^D
\hspace{6cm}\end{aligned}$$ $$\begin{aligned}
&=&\mu_1\mu_2\psi^C_{\mu_1}\phi_{\mu_2}^D\sum_{r=1}^n\sum_{r\neq
s\in{\overline}{1,n}} \frac{z_r+z_r^{-1}}{z_r+z_r^{-1}-z_s-z_s^{-1}}\\
&=&\frac{n(n-1)}{2}\mu_1\mu_2\psi^C_{\mu_1}
\phi_{\mu_2}^D.\hspace{10cm}(6.28)\end{aligned}$$ Set $$\phi_{\mu_1,\mu_2}^C=\psi^C_{\mu_1}\phi_{\mu_2}^D=\left[\prod_{r=1}^n(z_r-z_r^{-1})^{\mu_1}
\right]\prod_{1\leq i<j\leq n}(z_i+z_i^{-1}-z_j-z_j^{-1})^{\mu_2}\eqno(6.29)$$ for $\mu_1,\mu_2\in{\mathbb}{C}$. By Theorem 6.3 and (6.27)-(6.29), we have:

<span>**Theorem 6.4**</span>. <span>*The function $\phi^C_{\mu_1,\mu_2}$ satisfies: $$\begin{aligned}
& &\sum_{r=1}^n(z_r{\partial}_{z_r})^2(\phi^C_{\mu_1,\mu_2})+[4\mu_1(1-\mu_1)
\sum_{r=1}^n\frac{z_r^2}{(z_r^2-1)^2}\\ &
&+2\mu_2(1-\mu_2)\left(\sum_{1\leq i<j\leq n}
\frac{z_iz_j}{(z_i-z_j)^2}+\sum_{1\leq i<j\leq
n}\frac{z_iz_j}{(z_iz_j-1)^2}\right)] \phi^C_{\mu_1,\mu_2}\\
&=&\left(n\mu^2_1+n(n-1)\mu_1\mu_2+\frac{n(n-1)(2n-1)\mu_2^2}{6}\right)
\phi^C_{\mu_1,\mu_2},\hspace{4.4cm}(6.30)\end{aligned}$$ which is an Olshanesky-Perelomov equation*</span>.

Observe that ${\cal W}_{C_n}=\phi^C_{1,1}$. The Weyl function of $so(2n+1)$ is $${\cal W}_{B_n}=\left[\prod_{r=1}^n(z_r^{1/2}-z_r^{-1/2})\right]\prod_{1\leq i<j\leq n}
(z_i+z_i^{-1}-(z_j+z_j^{-1})),\eqno(6.31)$$ In terms of determinant, $$\begin{aligned}
& &{\cal W}_{B_n}=\\
& &\left|\begin{array}{ccc}z_1^{1/2}-z_1^{-1/2}&\cdots& z_n^{1/2}-z_n^{-1/2}\\ (z_1^{1/2}
-z_1^{-1/2})(z_1^{1/2}+z_1^{-1/2})^2&\cdots& (z_n^{1/2}-z_n^{-1/2})(z_n^{1/2}+z_n^{-1/2})^2\\
(z_1^{1/2}-z_1^{-1/2})(z_1^{1/2}+z_1^{-1/2})^4&\cdots&
(z_n^{1/2}-z_n^{-1/2})(z_n^{1/2} +z_n^{-1/2})^4\\
\vdots&\vdots&\vdots\\
(z_1^{1/2}-z_1^{-1/2})(z_1^{1/2}+z_1^{-1/2})^{2(n-1)} &\cdots&
(z_n^{1/2}-z_n^{-1/2})(z_n^{1/2}+z_n^{-1/2})^{2(n-1)}\end{array}\right|.
\hspace{0.6cm}(6.32)\end{aligned}$$ Moreover, $$\begin{aligned}
\qquad & &(z_r{\partial}_{z_r})^2[(z_r^{1/2}-z_r^{-1/2})(z_r^{1/2}+z_r^{-1/2})^{2k}]
\\ &=&\left(k^2+k+\frac{1}{4}\right)(z_r^{1/2}-z_r^{-1/2})(z_r^{1/2}+z_r^{-1/2})^{2k}\\
&
&+2k(2k-1)(z_r^{1/2}-z_r^{-1/2})(z_r^{1/2}+z_r^{-1/2})^{2(k-1)}.\hspace{4.9cm}(6.33)\end{aligned}$$ Thus $$(\sum_{i=1}^n(z_i{\partial}_{z_i})^2)({\cal W}_{B_n})=\frac{n(4n^2-1)}{12}{\cal W}_{B_n}\eqno(6.34)$$ by Lemma 6.1 and (6.33).

Set $$\psi^B_\mu=\prod_{r=1}^n(z_r^{1/2}-z_r^{-1/2})^{\mu}\qquad{\mbox{for}}\;\;\mu\in{\mathbb}{C}.\eqno(6.35)$$ Then $$\begin{aligned}
\sum_{r=1}^n(z_r{\partial}{z_r})^2(\psi^B_\mu)&=&\frac{\mu}{2}\psi^B_\mu
\sum_{r=1}^n\left[z_r{\partial}{z_r}\left(\frac{z_r^{1/2}+z_r^{-1/2}}{z_r^{1/2}-z_r^{-1/2}}\right)
+\frac{\mu}{2}\left(\frac{z_r^{1/2}+z_r^{-1/2}}{z_r^{1/2}-z_r^{-1/2}}\right)^2\right]\\ &=&
\frac{\mu}{4}\psi^B_\mu\sum_{r=1}^n\left[\frac{(z_r^{1/2}-z_r^{-1/2})^2-(z_r^{1/2}+
z_r^{-1/2})^2}{(z_r^{1/2}-z_r^{-1/2})^2}+\mu\frac{(z_r^{1/2}+z_r^{-1/2})^2}{(z_r^{1/2}
-z_r^{-1/2})^2}\right]\\  &=&\left[\frac{n\mu^2}{4}+\mu(\mu-1)\sum_{r=1}^n\frac{z_r}
{(z_r-1)^2}\right]\psi^B_\mu.\hspace{4.5cm}(6.36)\end{aligned}$$ Moreover, $$\begin{aligned}
\qquad& &\sum_{r=1}^n(z_r{\partial}{z_r})(\psi^B_{\mu_1})(z_r{\partial}{z_r})
(\phi_{\mu_2}^D)\\ &=&\frac{\mu_1\mu_2}{2}\sum_{r=1}^n\frac{z_r^{1/2}+z_r^{-1/2}}
{z_r^{1/2}-z_r^{-1/2}}\psi^B_{\mu_1}\sum_{r\neq s\in{\overline}{1,n}}\frac{z_r-z_r^{-1}}
{z_r+z_r^{-1}-z_s-z_s^{-1}}\phi_{\mu_2}^D\\ &=&\frac{\mu_1\mu_2}{2}\psi^B_{\mu_1}
\phi_{\mu_2}^D\sum_{r=1}^n\sum_{r\neq s\in{\overline}{1,n}}\frac{(z_r^{1/2}+z_r^{-1/2})^2}
{z_r+z_r^{-1}-z_s-z_s^{-1}}\\ &=&\frac{n(n-1)}{4}\mu_1\mu_2\psi^B_{\mu_1}\phi_{\mu_2}^D.
\hspace{9.2cm}(6.37)\end{aligned}$$ Set $$\phi_{\mu_1,\mu_2}^B=\psi^B_{\mu_1}\phi_{\mu_2}^D=\left[\prod_{r=1}^n(z_r^{1/2}
-z_r^{-1/2})^{\mu_1}\right]\prod_{1\leq i<j\leq n}(z_i+z_i^{-1}-z_j-z_j^{-1})^{\mu_2}
\eqno(6.38)$$ for $\mu_1,\mu_2\in{\mathbb}{C}$. By Theorem 6.3 and (6.36)-(6.38), we obtain:

<span>**Theorem 6.5**</span>. <span>*The function $\phi^B_{\mu_1,\mu_2}$ satisfies: $$\begin{aligned}
& &\sum_{r=1}^n(z_r{\partial}_{z_r})^2(\phi^B_{\mu_1,\mu_2})+
[\mu_1(1-\mu_1)\sum_{r=1}^n\frac{z_r}{(z_r-1)^2}\\ & &+2\mu_2(1-\mu_2)
\left(\sum_{1\leq i<j\leq n}\frac{z_iz_j}{(z_i-z_j)^2}+\sum_{1\leq i<j\leq n}\frac{z_iz_j}
{(z_iz_j-1)^2}\right)]\phi^C_{\mu_1,\mu_2}\\ &=&\left(\frac{n\mu^2_1}{4}+\frac{n(n-1)}{2}
\mu_1\mu_2+\frac{n(n-1)(2n-1)\mu_2^2}{6}\right)\phi^B_{\mu_1,\mu_2},\hspace{4.2cm}(6.39)\end{aligned}$$ which is an Olshanesky-Perelomov equation*</span>.

Olshanesky-Perelomov Equation of Type C
=======================================

In this section, we want to show that the trace functions of certain intertwining operators among $sp(2n)$-modules give rise to solutions of the equation (1.13) with $K_3=0$, following Etingof's idea in \[Ep\].

Recall the sympletic Lie algebra $sp(2n)$ in (4.1) and the notions $\{C_{r,s}\}$ in (4.8)-(4.10). The Casimier element $$\begin{aligned}
\qquad \Omega&=&\sum_{i,j=1}^nC_{i,j}C_{j,i}+\sum_{1\leq s<r\leq n}
(C_{n+r,s}C_{s,n+r}+C_{s,n+r}C_{n+r,s})\\ &
&+2\sum_{p=1}^n(C_{n+p,p}C_{p,n+p}+C_{p,n+p}
C_{n+p,p}),\hspace{6.2cm}(7.1)\end{aligned}$$ which is a quadratic central element in $U(sp(2n))$. Denote by $T^\ast$ the set of all linear functions (weights) on the toral Cartan subalgebra $T$ in (4.3). Let $$M=\bigoplus_{{\lambda}\in T^\ast}M^{\lambda}\eqno(7.2)$$ be any weight $sp(2n)$-module with the representation $\pi_{_M}$ such that $$\pi_{_M}(\Omega)=\mu \mbox{Id}_M\qquad\mbox{for some}\;\;\mu\in {\mathbb}{C},\eqno(7.3)$$ where $$M^{\lambda}=\{w\in M\mid h(w)={\lambda}(h)w\;{\mbox{for}}\;h\in T\}.\eqno(7.4)$$ In particular, we can take $M$ to be any highest weight module; say $M_{\lambda}$ Section 4. The module $M$ is not necessarily irreducible. We use the notations in (4.29) and (4.30). Denote the space of formal Laurent series in $\{x_1,x_2,...,x_n\}$ with the coefficients in $M$ by $${\tilde}{M}=\{\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast\mid w_n\in M\}.\eqno(7.5)$$ We extend the $\pi_{_M}$ to ${\tilde}{M}$ by $$\pi_{_M}(u)(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=\sum_{\vec i\in{\mathbb}{Z}^n}
\pi_{_M}(u)(w_{\vec i})x^{\vec i}x^\ast\qquad{\mbox{for}}\;\;u\in sp(2n).\eqno(7.6)$$ Moreover, we define another representation $\pi'$ of $sp(2n)$ on ${\tilde}{M}$ by $$\pi'(C_{p,q})(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=\sum_{\vec i\in{\mathbb}{Z}^n}
\left(C_{p,q}(w_{\vec i}x_p{\partial}_{x_q}(x^{\vec i}x^\ast)+\frac{{\delta}_{p,q}}{2}w_{\vec i}x^{\vec i}
x^\ast\right),\eqno(7.7)$$ $$\pi'(C_{p,n+q})(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=-\frac{1}{1+{\delta}_{p,q}}
\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x_px_qx^{\vec i}x^\ast,\eqno(7.8)$$ $$\pi'(C_{n+p,q})(\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}x^{\vec i}x^\ast)=
\frac{1}{1+{\delta}_{p,q}}\sum_{\vec i\in{\mathbb}{Z}^n}w_{\vec i}{\partial}_{x_p}{\partial}_{x_q}
(x^{\vec i}x^\ast)\eqno(7.9)$$ for $p,q\in{\overline}{1 n}$. Set $$\pi=\pi_{_M}+\pi'.\eqno(7.10)$$

Suppose that $\Phi:M{\rightarrow}{\tilde}{M}$ is a linear map such that $$\Phi(\pi_{_M}(u)(w))=\pi(u)\Phi(w)\eqno(7.11)$$ for $u\in sp(2n)$ and $w\in M$. View $\Phi$ as a function in $\{x_1,x_2,...,x_n\}$ taking value in the spaces of linear transformations on $M_{\lambda}$. Define the trace function: $${\cal E}=(x^\ast)^{-1}\mbox{tr}_M\Phi z_1^{\pi_{_M}(C_{1,1})} z_2^{\pi_{_M}(C_{2,2})}
\cdots  z_n^{\pi_{_M}(C_{n,n})}.\eqno(7.12)$$ We want to prove that the function $$\Psi={\cal W}_{C_n}{\cal E}\eqno(7.13)$$ satisfies the Olshanesky-Perelomov equation (1.13) with $K_3=0$. Note that (6.22) implies $$\begin{aligned}
z_i{\partial}_{z_i}({\cal W}_{C_n})&=&\sum_{i=1}^n\left[\frac{z_i+z_i^{-1}}{z_i
-z_i^{-1}}+\sum_{i\neq r\in{\overline}{1,n}}\frac{z_i-z_i^{-1}}{z_i+z_i^{-1}-z_r-z_r^{-1}}\right]
{\cal W}_{C_n}\\ &=&\sum_{i=1}^n\left[\frac{z_i^2+1}{z_i^2-1}+\sum_{i\neq r\in{\overline}{1,n}}
\left(\frac{z_i}{z_i-z_r}+\frac{1}{z_iz_r-1}\right)\right]{\cal W}_{C_n}.\hspace{3cm}
(7.14)\end{aligned}$$

Set $${\cal E}^{i_1,i_2}=(x^\ast)^{-1}\mbox{tr}_M\Phi \pi_{_M}( C_{i_1,i_2}C_{i_2,i_1})
z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\eqno(7.15)$$ for $i_1,i_2\in{\overline}{1,n}$. Then $${\cal E}^{i,i}=(z_i{\partial}_{z_i})^2{\cal E}\eqno(7.16)$$ and for $i_1\neq i_2$, $$\begin{aligned}
 x^{\ast}{\cal E}^{i_1,i_2}&=&\mbox{tr}_M\pi(C_{i_1,i_2})\Phi \pi_{_M}
(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}
\\ &=& \mbox{tr}_M\pi'(C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}
z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}\\ &
&+\mbox{tr}_M\pi_{_M}(C_{i_1,i_2}) \Phi
\pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M} (C_{n,n})}\\ &=&\mbox{tr}_M\pi'(C_{i_1,i_2})\Phi
\pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}
z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}\\ &
&+\mbox{tr}_M \Phi
\pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\pi_{_M}(C_{i_1,i_2})\\
&=&\mbox{tr}_M\pi'(C_{i_1,i_2}) \Phi
\pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\\ &
&+\frac{z_{i_1}}{z_{i_2}}\mbox{tr}_M\Phi \pi_{_M}(C_{i_2,i_1}
C_{i_1,i_2})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}\\
 &=& \mbox{tr}_M\pi'(C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}
 z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}\\ & &+\frac{z_{i_1}}{z_{i_2}}
 \mbox{tr}_M\Phi \pi_{_M}(C_{i_1,i_2}C_{i_2,i_1}+[C_{i_2,i_1},C_{i_1,i_2}])z_1^{\pi_{_M}
 (C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
 z_n^{\pi_{_M}(C_{n,n})}\hspace{3cm}\end{aligned}$$ $$\begin{aligned}
&=& \mbox{tr}_M\pi'
 (C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
  z_n^{\pi_{_M}(C_{n,n})}\\ & &+\frac{z_{i_1}}{z_{i_2}}\mbox{tr}_M\Phi \pi_{_M}(C_{i_1,i_2}
  C_{i_2,i_1}+C_{i_2,i_2}-C_{i_1,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
   z_n^{\pi_{_M}(C_{n,n})}\\ &=&
\mbox{tr}_M\pi'(C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}
(C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}+\frac{z_{i_1}}{z_{i_2}}x^\ast {\cal E}^{i_1,i_2}\\
 & &+\frac{z_{i_1}}{z_{i_2}}(z_{i_2}{\partial}_{z_{i_2}}-z_{i_1}{\partial}_{z_{i_1}})x^\ast {\cal E}.
 \hspace{9.2cm}(7.17)\end{aligned}$$ Thus $$\begin{aligned}
x^\ast{\cal E}^{i_1,i_2}&=&\frac{z_{i_2}}{z_{i_2}-z_{i_1}}\mbox{tr}_M
\pi'(C_{i_1,i_2})\Phi
\pi_{_M}(C_{i_2,i_1})z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}
\cdots  z_n^{\pi_{_M}(C_{n,n})}\\ &
&+x^\ast\frac{z_{i_1}}{z_{i_2}-z_{i_1}}(z_{i_2}
{\partial}_{z_{i_2}}-z_{i_1}{\partial}_{z_{i_1}}) {\cal
E}.\hspace{7cm}(7.18)\end{aligned}$$ Moreover, $$\begin{aligned}
& &\mbox{tr}_M\pi'(C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1})
z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\\ &=&
\mbox{tr}_M\pi'(C_{i_1,i_2})\pi(C_{i_2,i_1})\Phi
z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M} (C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\\ &=&
 \mbox{tr}_M\pi'(C_{i_1,i_2})\pi'(C_{i_2,i_1})\Phi z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}
 (C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}\\ & &+ \mbox{tr}_M\pi'(C_{i_1,i_2})\pi_{_M}
 (C_{i_2,i_1})\Phi z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}
 (C_{n,n})}\\ &=&x_{i_1}{\partial}_{x_{i_2}}x_{i_2}{\partial}_{x_{i_1}} (x^\ast {\cal E})+
 \mbox{tr}_M\pi'(C_{i_1,i_2})\Phi z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
 z_n^{\pi_{_M}(C_{n,n})}\pi_{_M}(C_{i_2,i_1})\\ &=&-\frac{1}{4}x^\ast {\cal E}+ \frac{z_{i_2}}
 {z_{i_1}}\mbox{tr}_M\pi'(C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1}) z_1^{\pi_{_M}(C_{1,1})}
 z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}(C_{n,n})}\hspace{2.1cm}(7.19)\end{aligned}$$ by (4.30). Hence $$\mbox{tr}_M\pi'(C_{i_1,i_2})\Phi \pi_{_M}(C_{i_2,i_1})
z_1^{\pi_{_M}(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}=-
\frac{z_{i_1}}{4(z_{i_1}-z_{i_2})}x^\ast{\cal E}.\eqno(7.20)$$ Substituting it to (7.18), we obtain $${\cal E}^{i_1,i_2}=\frac{z_{i_1}}{z_{i_2}-z_{i_1}}(z_{i_2}{\partial}_{z_{i_2}}-z_{i_1}
{\partial}_{z_{i_1}}) {\cal
E}+\frac{z_{i_1}z_{i_2}}{4(z_{i_1}-z_{i_2})^2}{\cal E}\qquad
{\mbox{for}}\;\;i_1,i_2\in{\overline}{1,n},\;i_1\neq i_2.\eqno(7.21)$$

Let $${\cal E}^{s,n+r}=(x^\ast)^{-1}\mbox{tr}_M\Phi \pi_{_M}( C_{s,n+r}C_{n+r,s})z_1^{\pi_{_M}
(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\eqno(7.22)$$ and $${\cal E}^{n+r,s}=(x^\ast)^{-1}\mbox{tr}_M\Phi \pi_{_M}(C_{n+r,s}C_{s,n+r})z_1^{\pi_{_M}
(C_{1,1})}z_2^{\pi_{_M}(C_{2,2})}\cdots
z_n^{\pi_{_M}(C_{n,n})}\eqno(7.23)$$ for $1\leq s\leq r\leq n$. By similar calculations as (7.17)-(7.21), we have $${\cal E}^{s,n+r}=\frac{z_rz_s}{z_rz_s-1}(z_r{\partial}_{z_r}+z_s{\partial}_{z_s}){\cal E}+\frac{z_rz_s}
{4(z_rz_s-1)^2}{\cal E}.\eqno(7.24)$$ $${\cal E}^{n+r,s}=\frac{1}{z_rz_s-1}(z_r{\partial}_{z_r}+z_s{\partial}_{z_s}){\cal E}+\frac{z_rz_s}
{4(z_rz_s-1)^2}{\cal E}\eqno(7.25)$$ for $1\leq s<r\leq n$ and $${\cal E}^{p,n+p}=\frac{z_p^2}{z_p^2-1}z_p{\partial}_{z_p}{\cal E}+\frac{3z_p^2}{16(z_p^2-1)^2}
{\cal E}.\eqno(7.26)$$ $${\cal E}^{n+p,p}=\frac{1}{z_p^2-1}z_p{\partial}_{z_p}{\cal E}+\frac{3z_p^2}{16(z_p^2-1)^2}
{\cal E}\eqno(7.27)$$ for $p\in{\overline}{1,n}$.

Now $$\begin{aligned}
\mu \Psi&=&{\cal W}_{C_n}(x^\ast)^{-1}\mbox{tr}_M\Phi \Omega z_1^{\pi_{_M}
(C_{1,1})} z_2^{\pi_{_M}(C_{2,2})}\cdots  z_n^{\pi_{_M}{C_{n,n}}}
\\ &=&{\cal W}_{C_n}[\sum_{i_1,i_2=1}^n{\cal E}^{i_1,i_2}+\sum_{1\leq s\leq r\leq n}
({\delta}_{r,s}+1)({\cal E}^{n+r,s}+{\cal E}^{s,n+r})]\\ &=& {\cal
W}_{C_n}[\sum_{i=1}^n(z_i{\partial}_{z_i})^2({\cal
E})+\sum_{i_1,i_2\in{\overline}{1,n}\:i_1\neq i_2}
\left(\frac{z_{i_1}}{z_{i_2}-z_{i_1}}(z_{i_2}{\partial}_{z_{i_2}}-z_{i_1}{\partial}_{z_{i_1}})
{\cal E} +\frac{z_{i_1}z_{i_2}}{4(z_{i_1}-z_{i_2})^2}{\cal
E}\right)\\ & &+\sum_{1\leq s<r\leq n}
\left(\frac{z_rz_s+1}{z_rz_s-1}(z_r{\partial}_{z_r}+z_s{\partial}_{z_s}){\cal
E}+\frac{z_rz_s}{2(z_rz_s-1)^2}{\cal E}\right)\\&
&+2\sum_{p=1}^n\left(\frac{z_p^2+1}{z_p^2-1}z_p{\partial}_{z_p}{\cal
E}+\frac{3z_p^2}{8(z_p^2-1)^2}{\cal E}\right)]\\
&=&\left[\sum_{1\leq i_1<i_2\leq
n}\frac{1}{2}\left(\frac{z_{i_1}z_{i_2}}{(z_{i_1}-z_{i_2})^2}+
\frac{z_{i_1}z_{i_2}}{(z_{i_1}z_{i_2}-1)^2}\right)+\frac{3}{4}\sum_{p=1}^n
\frac{z_p^2}{(z_p^2-1)^2}\right]\Psi\\
& &+ 2\sum_{r,s\in{\overline}{1,n};\;r\neq
s}\left(\frac{z_r}{z_r-z_s}+\frac{1}{z_rz_s-1}\right)z_r{\partial}_{z_r}{\cal
E}+2\sum_{p=1}^n\frac{z_p^2+1}{z_p^2-1}z_p{\partial}_{z_p}{\cal E}\\ &
&+{\cal W}_{C_n}\sum_{i=1}^n(z_i{\partial}_{z_i})^2({\cal E})\\&=&
\left[\sum_{1<i_1<i_2\leq
n}\frac{1}{2}\left(\frac{z_{i_1}z_{i_2}}{(z_{i_1}-z_{i_2})^2}+\frac{z_{i_1}z_{i_2}}
{(z_{i_1}z_{i_2}-1)^2}\right)+\frac{3}{4}\sum_{p=1}^n\frac{z_p^2}{(z_p^2-1)^2}\right]\Psi\\
& &+\sum_{i=1}^n[{\cal W}_{C_n}(z_i{\partial}_{z_i})^2({\cal
E})+2z_i{\partial}_{z_i}({\cal W}_{C_n})(z_i{\partial}_{z_i})({\cal E})]\\ &=&
\left[\sum_{1<i_1<i_2\leq
n}\frac{1}{2}\left(\frac{z_{i_1}z_{i_2}}{(z_{i_1}-z_{i_2})^2}+\frac{z_{i_1}z_{i_2}}
{(z_{i_1}z_{i_2}-1)^2}\right)+\frac{3}{4}\sum_{p=1}^n\frac{z_p^2}{(z_p^2-1)^2}\right]\Psi\\
&
&+\sum_{i=1}^n(z_i{\partial}_{z_i})^2(\Psi)-\sum_{i=1}^n(z_i{\partial}_{z_i})^2({\cal
W}_{C_n}){\cal E}\\ &=&\left[\sum_{1<i_1<i_2\leq
n}\frac{1}{2}\left(\frac{z_{i_1}z_{i_2}}{(z_{i_1}-z_{i_2})^2}+\frac{z_{i_1}z_{i_2}}
{(z_{i_1}z_{i_2}-1)^2}\right)+\frac{3}{4}\sum_{p=1}^n\frac{z_p^2}{(z_p^2-1)^2}\right]\Psi\\
&
&+\sum_{i=1}^n(z_i{\partial}_{z_i})^2(\Psi)-\frac{n(n+1)(2n+1)}{6}\psi\hspace{6.5cm}(7.28)\end{aligned}$$ by (6.25), (7.12)-(7.16), (7.21) and (7.24)-(7.27). Therefore, we have:

<span>**Theorem 7.1**</span>. <span>*The function $\Psi$ in (7.13) satisfies the following Olshanesky-Perelomov equation of type C*</span>: $$\begin{aligned}
& &\sum_{i=1}^n(z_i{\partial}_{z_i})^2(\Psi)+\left[\sum_{1<i_1<i_2\leq n}
\frac{1}{2}\left(\frac{z_{i_1}z_{i_2}}{(z_{i_1}-z_{i_2})^2}+\frac{z_{i_1}z_{i_2}}
{(z_{i_1}z_{i_2}-1)^2}\right)+\frac{3}{4}\sum_{p=1}^n\frac{z_p^2}{(z_p^2-1)^2}\right]\Psi\\
 &=&\left(\mu+\frac{n(n+1)(2n+1)}{6}\right)\Psi.\hspace{8.7cm}(7.29)\end{aligned}$$

In particular, when $M=M_{\lambda}$ in (4.23), $$\Psi={\cal W}_{C_n}E_C=
z_n^{{\lambda}_n+1}\left[\prod_{i=1}^{n-1}z_i^{n-i+1/2}\right]{\cal
X}_C(1/2,...,1/2;-{\lambda}_n)\{\xi^A_{r_2,r_1},\xi^C_{n+s_2,s_1}\}\eqno(7.30)$$ (cf. (4.115)) satisfies the above equation with $$\mu= \sum_{i=0}^{n-1}\left({\lambda}_n-\frac{i}{2}\right)\left({\lambda}_n+\frac{3i}{2}+2\right).\eqno(7.31)$$

<span>**References**</span>

<span>\[AAR\]</span>

:   G. Andrews, R. Askey and R. Roy, <span>*Special Functions*</span>, Cambridge University Press, 1999.

<span>\[BO\]</span>

:   R. Beerends and E. Opdam, Certain hypergeometric series related to the root system $BC$, <span>*Trans. Amer. Math. Soc.* </span> <span>**119**</span> (1993), 581-609.

<span>\[BH\]</span>

:   F. Beukers and G. Heckman, Monodromy for a hypergeometric function $_nF_{n-1}$, <span>*Invent. Math.*</span> <span>**95**</span> (1989), no. 2, 325-354.

<span>\[BL\]</span>

:   D. Briten and F. Lemire, A classification of simple Lie modules having a 1-dimensional weight modules, <span>*Trans. Amer. Math. Soc.*</span> <span>**299**</span> (1987), no.2, 683-697.

<span>\[C\]</span>

:   F. Calogero, Solution of the one-dimensional $n$-body problem with quadratic and /or inversely quadratic pair potentials, <span>*J. Math. Phys.*</span> <span>**12**</span> (1971), 419-432.

<span>\[Ep\]</span>

:   P. Etingof, Quantum integrable systems and representations of Lie algebras, <span>*J. Math. Phys.*</span> <span>**36**</span> (1995), no. 6,2637-2651.

<span>\[Eh\]</span>

:   H. Exton, Multiple hypergeometric functions and applications, Halsted Pree (John Wiley & Sons, Inc,), New York, 1976.

<span>\[F\]</span>

:   S. L. Fernando, Simple weight modules of complex reductive Lie algebras, <span>*Ph. D. Thesis, Univ. of Wisconsin at Madison*</span>, 1983.

<span>\[GG\]</span>

:   I. M. Gel'fand and M. I. Graev, GG-functions and their relation to general hypergeometric functions, <span>*Russian Math. Surveys*</span> <span>**52**</span>(1997), no. 4, 639-684.

<span>\[Hg1\]</span>

:   G. Heckman, Root systems and hypergeometric functions II, <span>*Compositio. Math.*</span> <span>**64**</span> (1987), 353-373.

<span>\[Hg2\]</span>

:   G. Heckman, Heck algebras and hypergeometric functions, <span>*Invent. Math.*</span> <span>**100**</span> (1990), 403-417.

<span>\[Hg3\]</span>

:   G. Heckman, An elementary approach to the hypergeometric shift operators of Opdam, <span>*Invent. Math.*</span> <span>**103**</span> (1990), 341-350.

<span>\[HO\]</span>

:   G. Heckman and E. Opdam, Root systems and hypergeometric functions I, <span>*Compositio. Math.*</span> <span>**64**</span> (1987), 329-352.

<span>\[Hj\]</span>

:   J. E. Humphreys, <span>*Introduction to Lie Algebras and Representation Theory*</span>, Springer-Verlag New York Inc., 1972.

<span>\[J\]</span>

:   J. C. Jantzen, Kontravariante formen auf induzierten Darstellungen habeinfacher Lie-algebren, <span>*Math. Ann.*</span> <span>**226**</span> (1977), no. 1, 53-65.

<span>\[K\]</span>

:   D. Kazhdan, B. Konstant and S. Steinberg, Hamiltonian group actions and dynamical system of Calogero type, <span>*Commun. Pure Appl. Math.*</span> <span>**31**</span> (1978), 481-507.

<span>\[M\]</span>

:   M. Marvan, Reducibility of zero curvature representations with application to recursion operators, <span>*arXiv:nlin.SI/0306006*</span>.

<span>\[NFP\]</span>

:   J. F. Núnez, W. G. Fuertes and A. M. Perelonov, Some results on the eigenfunctions of the quantum trigonometric Calogero-Sutherland model related to the Lie algebra $D_4$, <span>*arXiv:math-ph/0305012*</span>, 7 May 2003.

<span>\[O1\]</span>

:   E. Opdam, Root systems and hypergeometric functions III, <span>*Compositio. Math.*</span> <span>**67**</span> (1988), 21-49.

<span>\[O2\]</span>

:   E. Opdam, Root systems and hypergeometric functions IVI, <span>*Compositio. Math.*</span> <span>**67**</span> (1988), 191-209.

<span>\[O3\]</span>

:   E. Opdam, Some applications of hypergeometric shift operators, <span>*Invent. Math.*</span> <span>**98**</span> (1989), 1-18.

<span>\[O4\]</span>

:   E. Opdam, An analogue of the Gauss summation formula for hypergeometric functions related to root systems, <span>*Math. Z.*</span> <span>**212**</span> (1993), 313-336.

<span>\[O5\]</span>

:   E. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, <span>*Acta Math.*</span> <span>**175**</span> (1995), 75-121.

<span>\[O6\]</span>

:   E. Opdam, Cuspital hypergeometric functions, <span>*Methods Appl. Anal.*</span> <span>**6**</span> (1999), 67-80.

<span>\[OP\]</span>

:   M. Olslanetsky and A. Perelomov, Completely integrable Hamiltonian systems connected with semisimple Lie algebras, <span>*Invent. Math.*</span> <span>**37**</span> (1976), 93-108.

<span>\[S\]</span>

:   B. Sutherland, Exact results for a quantum many-body problem in one-dimension, <span>*Phys. Rev. A*</span> <span>**5**</span> (1972), 1372-1376.

<span>\[V1\]</span>

:   D,-N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, <span>*thesis, Yale University,*</span> 1966.

<span>\[V2\]</span>

:   D,-N. Verma, Structure of certain induced representations of complex semisimple Lie algebras, <span>*Bull. Amer. Math. Soc.*</span> <span>**74**</span>(1968), 160-166.

<span>\[X\]</span>

:   X. Xu, Differential equations for singular vectors of $sl(n)$, <span>*Preprint*</span>.

[^1]: 2000 Mathematics Subject Classification: 17B80, 33D67, 35Q58

[^2]: Research supported by CNSF Grant 10371121

{'source': 'arXiv_out/math0411147'}