数理解析研究所講究録 1227 巻 2001 年 14-60

14Density matrices and LR transforms(Genesis of Orthogonal Functions)Kazuhiko AOMOTO(Graduate School of Mathematics, Nagoya University)
April 3, 2001

1Introduction

Th.e two kinds of Chebyschev polynomials$\frac{\mathrm{s}\mathrm{i}n(n+1)\theta}{\sin\theta}$$T_{n}(\cos\theta)=\cos n\theta$

are linearly related to each other by the formulat

and $U_{n}(\cos\theta)=$$\cos n\theta=\frac{1}{2}\{\frac{\sin(n+1)\theta}{\sin\theta}-\cdot\frac{\sin(n,-1)\theta}{\sin\theta}\}$Both polynomials satisfy the difference equations$xu_{n}= \frac{1}{2}(u_{n+1}+u_{n-1})$

This is asimplest case of -transforms associated with difference operatorsfor orthogonal functions.
Asystem of orthogonal functions are intimately related with eigenfunctions for aself-adjoint operator through density matrices. Once afamilyof self-adjoint operators are given, we can discuss the interplay among LRtransforms of self-adjoint operators, linear transforms of density matrices
and connection relations between two system of eigen-functions for the operators. This mechanism enables us to give anew orthogonal system from theprevious one, and so on.
Let be an infinite real -diagonal matrix
which defines
abounded self-adjoint operator on. There exist the spectral kernels$d(n, m|\lambda)$ which are the Stieltjes measures onsuch that$\mathrm{L}\mathrm{R}$$A$$\mathrm{t}\mathrm{r}\mathrm{i}$$(a_{n,m})_{n,m=-\infty}^{\infty}$$l^{2}(\mathrm{Z})$$\mathrm{R}$15$\delta_{n,m}=\int_{-\infty}^{\infty}d(n, m;\lambda)$(1.1)$a_{n,m}= \int_{-\infty}^{\infty}\lambda d(n, m;\lambda)$(1.2)

The eigenfunction expansion forusing generalized eigenfunctions$A$

is an expression of $d(n, m;\lambda)$ , byof satisfying$A$$\psi^{(\epsilon)}(n;\lambda)(\epsilon=\pm)$(1.3)$A\psi^{(\epsilon)}(n;\lambda)=\lambda\psi^{(\epsilon)}(n;\lambda)$and Stieltjes measures called density matrices$d\rho_{\epsilon,\epsilon’}(\lambda)$, as$d(n, m; \lambda)=\sum_{\epsilon=\pm,\epsilon’=\pm}\psi^{(\epsilon)}(n;\lambda)\psi^{(\epsilon’)}(n\iota;\lambda)d\rho_{\epsilon,\epsilon’}(\lambda)$(1.4)

be apositive continuous function such that $f(A)$ defines aposLet.
itive definite operator on
Assume that there exists aGauss decomposition of $f(A)$ of the following
type$f(\lambda)$$l^{2}(\mathrm{Z})$(1.5)$f(A)=B_{-}\cdot B_{+}$) denotes an upper triangular(or $B_{-}={}^{t}B_{+}$ the transpose of
where
are also well-defined.
(or lower triangular) matrix such that the inverses
Then the $LR$ transform of can be defined as follows.
$B_{+}$$B_{+}$$B_{\pm}^{-1}$$A$$Aarrow A’=B_{-}^{-1}\cdot A\cdot B_{-}=B_{+}\cdot A\cdot B_{+}^{-1}$(1.6)

In this note we show that this transform is equivalent to acertain linear orand evaluate it explicitly
projective transform of the density matrices
in the following four cases(1) Orthogonal polynomials in asingle variable(2) Inverse scattering case(3) Periodic case(4) Orthogonal polynomials in multi-variables
respectively.
$d\rho_{\epsilon,\epsilon’}(\lambda)$

This note has been written in collaboration with Dr.Masahiko Ito. Especially the computations for proving Proposition 8are mostly due to him

162

Orthogonal polynomials in asingle variablewith infinite increments and whose
We consider aStieltjes measure
support is contained in the finite interval $[a, b]$ $(a<b)$ in R. There exist theunique orthonormal polynomials in A
$d\rho(\lambda)$$p_{0}(\lambda)$,$p_{1}(\lambda)$,$p_{2}(\lambda)$,$\ldots$(we put $p_{-1}(\lambda)=0$ ) such that they satisfy$p_{n}(\lambda)=k_{n}\lambda^{n}+$(lower degree terms)$k_{n}^{n}>0$(2. 1)(2.2)$\int_{a}^{b}p_{n}(\lambda)p_{m}(\lambda)d\rho(\lambda)=\delta_{n,m}$

The three term recurrence equations hold$\lambda p_{n}(\lambda)=b_{n-1}p_{n-1}(\lambda)+a_{n}p_{n}(\lambda)+b_{7\iota}p_{n+1}(\lambda)(n\geq 0)$

Let
that

$A$

denote the corresponding$a_{n,n}=a_{n}$$\mathrm{t}\mathrm{r}\mathrm{i}$-diagonal matrix, $a_{n,n+1}=a_{n+1,n}=b_{n}$$(a_{n,m})_{n,m=-\infty}^{\infty}$

(2.3)
such

(2.4)

$n\geq 0$

The matrix defines aself-adjoint operator on. has the spectral
decomposition (1.1), (1.2) where $d(n, m;\lambda)$ is represented simply by$A$$l^{2}(\mathrm{Z}_{\geq 0})$

$A$

(2.3)

$d(n, m; \lambda)=p_{n}(\lambda)p_{m}(\lambda)d\rho(\lambda)$

Let $f(x)$ be apositive continuous function on $[a, b]$ . Then $f(A)$ and$f(A)^{-1}$ define bounded self-adjoint operators. There exist the unique upper
triangular and lower triangular matrices
andwith positive diagonal
elements satisfying (1.5). All
are bounded operators.
andThe $LR$ transform of associated with the function
is defined by$A’$correspondence- (1.6). is again a -diagonal self-adjoint operator on$B_{+}$$B_{\pm}$$B_{\pm}^{-1}$$A$$f(\lambda)$$\mathrm{t}\mathrm{r}\mathrm{i}$$\mathrm{t}\mathrm{h},\mathrm{e}$$B_{-}$

17
Y.Nakamura and Y.Kodama, and also V.Spiridonov and A.Zhedanov
have investigated $LR$-transforms associated with finite matrices and orthogonal polynomials (see [23],[24],[30]). Here we want to relate them to linear.
(or projective) transforms of density matrices
In section 7-8 we extend-transforms to the case of orthogonal polyn0mials in multi-variables. In the final section we shall obtain explicit formulaefor-transforms associated with Koornwinder polynomials.
$d\rho(\lambda)$$\mathrm{L}\mathrm{R}$$\mathrm{L}\mathrm{R}$Proposition 1Letbe the density corresponding to the operator $A’$ .
The $LR$ transform (1.6) is equivalent to the linear correpondence$d\rho’(\lambda)$(2.6)$d\rho’(\lambda)=f(\lambda)d\rho(\lambda)$

If$d\rho(\lambda)$

and$d\rho’(\lambda)$

are nor malized such that
(2.7)$\int_{a}^{b}d\rho(\lambda)=\int_{a}^{b}d\rho’(\lambda)=1$

then (2.6) shoud be modified as(2.8)$d \rho(\lambda)arrow d\rho’(\lambda)=\frac{f(\lambda)d\rho(\lambda)}{\int_{a}^{b}f(\lambda)d\rho(\lambda)}$In fact, (2.6) implies the formulae$(f(A))_{n,n},= \int_{-\infty}^{\infty}p_{n}(/\backslash )p_{?n}(\lambda)d\rho’(\lambda)$

Let$\{p_{n}’(\lambda)\}$$d\rho’(\lambda)$.

$p_{n}(\lambda)$

be the orthonormal polynomials with respect to the density
can be expressed uniquely as alinear combination of$p_{m}’(\lambda)$$p_{n}( \lambda)=\sum_{m=0}^{n}b_{m,n}p_{m}’(\lambda)$Let
be the upper triangular matrix(2.9). On the other hand$B_{+}$(2.9)$(b_{n,?n})_{n,m=0}^{\infty}$(2.10). Then (1.5) holds from18

(2.11)$(A’)_{n,m}= \int_{-\infty}^{\infty}\lambda p_{n}’(\lambda)p_{m}’(\lambda)d\rho’(\lambda)$$\mathrm{R}\mathrm{o}\mathrm{m}$$(2.9)-(2.11)$ , we deduce (1.6).

In particular, if is itself positive definite andthe original Rutishauser’s $LR$ algorithm.
$A$$f(\lambda)=\lambda$

Examples 1. Jacobi polynomials.
Leton [-1, 1], forpolynomials
are defined by the equations$d\rho(\lambda)=(1-\lambda)^{\alpha}(1+\lambda)^{\beta}d\lambda$$\alpha$, (1.6) reduces to, $\beta>-1$ . The Jacobi$P_{n}^{(\alpha,\beta)}(\lambda)$$(1- \lambda)^{\alpha}(1+\lambda)^{\beta}P_{n}^{(\alpha,\beta)}(\lambda)=\frac{(-1)^{n}}{2^{n}n!}(\frac{d}{d\lambda})^{n}\{(1-\lambda)^{\alpha+n}(1+\lambda)^{\beta+n}\}$(2.12)Then$P_{n}^{(\alpha\beta)}(\lambda)=l_{n}^{(\alpha,\beta)}\lambda^{n}+\cdots$$l_{n}^{(\alpha\beta)}=2^{-n} \frac{\Gamma(2n+\alpha+\beta+1)}{\Gamma(n+1)\Gamma(n+\alpha+\beta+1)}$

and$\int_{-1}^{1}(1-\lambda)^{\alpha}(1+\lambda)^{\beta}P_{n}^{(\alpha,\beta)}(\lambda)P_{m}^{(\alpha,\beta)}(\lambda)d\lambda=0$,$n\neq m$$\int_{-1}^{1}(1-\lambda)^{\alpha}(1+\lambda)^{\beta}\{P_{n}^{(\alpha,\beta)}(\lambda)\}^{2}d\lambda=h_{n}^{(\alpha,\beta)}$$h_{n}^{(\alpha,\beta\rangle}= \frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+1)\Gamma(n+\alpha+\beta+1)}$

The reccurence equations for$P_{n}^{\alpha,\beta}(x)$are as follows.

$2(n+1)(n+1+\alpha+\beta)(2n+\alpha+\beta)P_{n+1}^{(\iota,\beta)}‘(\lambda)$$=$$(2n+\alpha+\beta+1)\{(2n+\alpha+\beta+2)(2n+\alpha+\beta)\lambda+\alpha^{2}-\beta^{2}\}P_{n}^{(\alpha.\beta)}(\lambda)$$-2(n+\alpha+1)(n+\beta+1)(2n+\alpha+\beta+2)P_{n-1}^{(\alpha,\beta)}(\lambda)$(2.13)

19$p_{n}(\lambda)=\{h_{n}^{(\alpha,\beta)}\}^{-\frac{1}{2}}P_{n}^{(\alpha,\beta)}(\lambda)$. We denoteis the orthonormal polynomials with respect tothenderived from (2.13).
by the -diagonal operator on
induces the transform of the densities
The shift$d\rho(\lambda)$$p_{n}(\lambda)$$A$$l^{2}(\mathrm{Z})_{\geq 0}$$\mathrm{t}\mathrm{r}\mathrm{i}$$\alphaarrow\alpha+1$(2.14)$d\rho(\lambda)arrow d\rho’(\lambda)=(1-\lambda)d\rho(\lambda)$

Since $1-A$ is positive definite, the Gauss decomposition(2.15)$1-A=B_{-}\cdot B_{+}$

is uniquely determined. Likewise we have
(2.16)$1+A=B_{-}\cdot B_{+}$

These are Christoffel-Darboux tranforms of contiguity relation.
In fact, if we put$\psi_{n}(\alpha, \beta)=\frac{1}{l_{n}^{(\alpha,\beta)}}P_{n}^{(\alpha,\beta)}(\lambda)=\lambda^{n}+\frac{n(\alpha-\beta)}{2n+\alpha+\beta}\lambda^{n-1}+\cdots$

then$\psi_{n}(\alpha, \beta)=\psi_{n}(\alpha+1, \beta)+v_{n}\psi_{n-1}(\alpha+1, \beta)$$v_{n}=- \frac{2n(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}$(2.17)

arealthoughare not bounded onMore exactly saying,as follows.
bounded. We must modify the operators
consisting of sequences $u=$
We denote by 7{ the Hilbert space. We define,
etc with the inner productanother Hilbert space Ho, the closed linear subspace spanned by $B_{+}u’(u’\in$.
is isomorphic to the Hilbert space consisting of the sequences$u=(u_{n})_{n\geq 0}$ such that $((1-A)^{-1}u, u)<\infty$ .
is abounded operator from$l^{2}(\mathrm{Z}),\cdot$$B_{\pm}^{-1}$$B_{\pm}$$l^{2}(\mathrm{Z}_{\geq 0})$$(u_{n})_{n=0}^{\infty}$$l^{2}(\mathrm{Z}_{\geq 0})$$v=(v_{n})_{n=0}^{\infty}$$(u, v)= \sum_{n=0}^{\infty}u_{n}\overline{v_{n}}$$\mathcal{H}_{0}$$B_{+}^{-1}$$B_{\pm}$

20
to 7-?, so thatis bounded as alinear mapping fromwhich is extendable to abounded operator on 7-?.
$?\# 0$$B.AB\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{\mathit{1}}$$?^{\ovalbox{\tt\small REJECT}}?0$to$7^{\ovalbox{\tt\small REJECT}}\langle$

Example 2. Askey-Wilson polynomials (see [5]).
Let be the real modulus such that $0<q<1$ , and , , , be real
numbers. Askey-Wilson polynomials are defined by using the basic hypergeometric series of order $m$$q$$c_{1}$$\mathrm{c}2$$c_{3}$$m \varphi_{m-1}(b_{1},\cdots,b_{m-1}a_{1},\cdots,a_{m} ; \lambda)=\sum_{\nu=0}^{\infty}.\cdot\frac{(a_{1},q)_{\nu}\cdots(a_{m},q)_{\nu}}{(b_{1},q)_{\nu}\cdots(b_{n-1}\cdot q)_{\nu}(q,q)_{\nu}},,\cdot.\lambda^{\nu}$$c_{4}$(2.18)

as$p_{n}(\lambda;c_{1}, c_{2}, c_{3}, c_{4})$$=c_{1}^{-n}(c_{1}c_{2};q)_{n}\cdot(c_{1}c_{3}; q)_{n}\cdot(c_{1}c_{4};q)_{n4}.\varphi_{3}(c_{1}c_{2},c_{3}c_{4},c_{1}c_{4}q^{-n},q^{n-1}c_{1}c_{2}c_{3}c_{4}, c_{1}e^{i\theta}, c_{1}e^{-i\theta} ; q)$$=l_{n}\lambda^{n}+\cdots$(2.19)$(l_{n}=2^{n}(c_{1}c_{2}c_{3}c_{4}q^{n};q)_{n})$where $\lambda=cosO$ . The weight function$w(\lambda)(d\rho(\lambda)=$is given by$w( \lambda)=.\frac{\Pi_{k=0}^{\infty}(1-2(2\lambda^{2}-1)q^{k}+q^{2k})}{h(\lambda,c_{1})h(\lambda,c_{2})h(\lambda,c_{3})h(\lambda,c_{4})}$(2.20)$h( \lambda, a)=\prod_{k=0}^{\infty}(1-2a\lambda q^{k}+q^{2k}a^{2})=(ae^{i\theta};q)_{\infty}(ae^{-i\theta};q)_{\infty}$(2.21).

where

Then the orthogonality relations are$\frac{1}{2\pi}\int_{-1}^{1}p_{n}(\lambda;c_{1}, c_{2}, c_{3}, c_{4})p_{m}(\lambda;c_{1}, c_{2}, c_{3}, c_{4})\frac{w(\lambda)}{\sqrt{1-\lambda^{2}}}d\lambda=\delta_{n,m}h_{n}$(2.22)

21h.

$(c_{r}c_{2}c_{3}c_{4Cl^{2n_{\ovalbox{\tt\small REJECT}}}}$$(j)_{oo}(c_{S}c_{2}c_{3}c_{4Cl^{n}}1\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}7)_{\mathrm{o}\mathrm{o}}(\ovalbox{\tt\small REJECT}/^{\mathrm{n}+1}:\ovalbox{\tt\small REJECT} 7)\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}(c_{\ovalbox{\tt\small REJECT}^{C}2Ci^{n}\ovalbox{\tt\small REJECT}(7)\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{1}}$(2.23)$\ovalbox{\tt\small REJECT}$$(c_{\mathrm{i}}c_{3}\mathrm{c}\mathrm{y}^{\mathrm{n}}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}7)_{\mathrm{D}\mathrm{O}}(c_{1}c_{4}q^{n}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{j})_{\mathrm{Q}\mathrm{Q}}(\mathrm{c}_{2}c_{3}q^{\mathrm{n}}\ovalbox{\tt\small REJECT} q)_{\mathrm{o}\mathrm{o}}(C_{2}C_{4}(\mathrm{j}"\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{j})_{\mathrm{Q}\mathrm{Q}}(c_{3}c_{4}q^{n}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{j})_{\mathrm{o}\mathrm{o}}$asThe three term recurrence relations for$p_{n}(\lambda;c_{1}, c_{2}, c_{3}, c_{4})$axe expressed(2.24)$2\lambda p_{n}(\lambda)=b_{n-1}p_{n-1}(\lambda)+a_{n}p_{n}(\lambda)+b_{n}’p_{n+1}(\lambda)$$b_{n-1}=(1-q^{n})(1-c_{1}c_{2}q^{n-1})(1-c_{1}c_{3}q^{n-1})(1-c_{1}c_{4}q^{n-1})$$\cross\frac{(1-c_{2}c_{3}q^{n-1})(1-c_{2}c_{4}q^{\tau\iota.-1})(1-c_{3}c_{4}q^{n-1})}{(1-cq^{2n-2})(1-cq^{2n-1})}$,$b_{n}’= \frac{1-cq^{n-1}}{(1-cq^{2\tau\iota-1})(1-cq^{2n})}$,$a_{n}= \frac{q^{n-1}[(1+cq^{2n-1})(sq+s’c)-q^{n-1}(1+q)(s+s’q)c]}{(1-cq^{2n-2})(1-cq^{2n})}$$(s=c_{1}+c_{2}+c_{3}+c_{4}, s’=c_{1}^{-1}+c_{2}^{-1}+c_{3}^{-1}+c_{4}^{-1}, c=c_{1}c_{2}c_{3}c_{4})$$d\rho(\lambda)$$T_{1}$

multiplies

depends on:$c_{1}$,,,$c_{4}$.

. In fact each shift: $c_{2}arrow c_{2}q;T_{3}$ : $c_{3}arrow c_{3}q;T_{4}$ :$c_{4}arrow c_{4}q$(2.25), $1+c_{2}^{2}-2c_{2}\lambda$ , $1+c_{3}^{2}-2c_{3}\lambda$ , $1+c_{4}^{2}-2c_{4}\lambda$(2.26)$c_{1}arrow c_{1}q;T_{2}$$w(\lambda)$$c_{2}$$\mathrm{c}3$by$1+c_{1}^{2}-2c_{1}\lambda$

times respectively.
The corresponding $LR$ transforms of
positions of each positive operator$A$

are defined as the Gauss decom-$1+A^{2}-2c_{1}A>0,1+A^{2}-2c_{2}A>0,1+A\underline{.)}-2c_{3}A>0,1+A^{2}-2c_{4}A>0$

Put$\psi_{n}(\lambda;c_{1}, c_{2}, c_{3}, c_{4})=\frac{1}{l_{n}}p_{n}(\lambda;c_{1}, c_{2}, c_{3}, c_{4})$(2.27)

22
then, as for T. for example, the transform B. is equivalent to the following
contiguity relation$\psi_{n}(x;c_{1}, c_{2}, c_{3}, c_{4})=\psi_{n}(x;c_{1}q, c_{2}, c_{3}, c_{4})+v_{n}\psi_{n-1}(x;c_{1}q, c_{2}, c_{3}, c_{4})$$v_{n}=- \frac{2(1-q)c_{1}}{(1-aq^{2n-2})(1-aq^{2n-1})(1-c_{2}c_{3}q^{n-1})(1-c_{2}c_{4}q^{n-1})(1-c_{3}c_{4}q^{n-1}}$

likewise for$T_{2}$, T3)$T_{4}$.

3Inverse scattering) Application of H.Flaschka
theory$A$be a-diagonal matrix which defines abounded self-adjoint operator on$\mathrm{t}\mathrm{r}\mathrm{i}$$\mathcal{H}=l^{2}(\mathrm{Z})$.

We put $a_{n,n}=a_{n}$ andthe following condition(C)$a_{n,n+1}=a_{n+1,n}=b_{n}\mathrm{f}\mathrm{o}\mathrm{r}-\infty<n<\infty$

and assume(3.1)$n= \sum_{-\infty}^{\infty}|a_{n}||n|<\infty,\sum_{n=-\infty}^{\infty}|b_{n}-\frac{1}{2}||n|<\infty$

The inverse scattering theory for the difference operator was developed.
by H.Flaschka (see [10],[32]). We put the spectral parameter
solutions in the sense, then the Jost solutions
Ifof S.Elaydi [9] are uniquely determined as the eigenfunctions (1.3) with theasymptotic behaviours$A$$z= \frac{1}{2}(\zeta+\zeta^{-1})$$|\zeta|\leq 1$$\psi^{\pm}(n;z)(\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{l}$$)$$\psi^{\pm}(n;z)\vee\zeta^{\pm n}\wedge$

The connection relations between(3.2)$narrow\pm\infty$$\psi^{\pm}(n.;z)$

are
(3.3)

$\psi^{-}(n;z)=\alpha(z)\tilde{\psi}^{\pm}(n;z)$ $+\beta(z)\psi^{\pm}(n;z)$when $(|\zeta|=$where
are defined to be the conjugates of,1) is replaced by.
can be holomorphically extended to the.
domain$\psi^{\pm}(n;z)$$\tilde{\psi}^{\pm}(n;z)$$\zeta^{-1}$$|\zeta|\leq’1$$\alpha(z)$$\beta(z)$

$($

23
The Wronskian and the reflection coefficients are expressed as$R(z)= \frac{\beta(z)}{\alpha(z)}$(3.4)$W( \psi_{+}, \psi_{-})=\frac{1}{2}(\zeta^{-1}-\zeta)\alpha(z)$(3.5)

respectively.
,
do exist. Moreover,,
For A $\in[-1,1]$ ,has afinite number of simple poles , $k=1,2,3$ , , such that.
Under this circumstance, it holds the following two expansion formulae
which are equivalent to each other.
$\psi^{\pm}(n;\lambda+\mathrm{i}\mathrm{O})$$\alpha(\lambda+i0)$$\alpha(z)$$\beta(\lambda+i0)$$\lambda_{k}$$\ldots$$s$$|\lambda_{k}|>$$1$

Proposition 2(1)$d(n, m;\lambda)$

where$= \frac{\chi_{[-1,1]}(\lambda)d\lambda}{2\pi\sqrt{1-\lambda^{2}}|\alpha(\lambda+i0)|^{2}}\{’\psi^{+}(n;\lambda+i0)\overline{\psi^{+}(m,\cdot\lambda+i0)}$$+$$\psi^{-}(n;\lambda+i0)\overline{\psi^{-}(m\cdot,\lambda+i\mathrm{O})}\}$$+$$. \sum_{k=1}^{s}\psi^{+}(n;\lambda_{k})\psi^{+}(m;\lambda_{k})c_{k}^{2}\delta(\lambda-\lambda_{k})d\lambda$$c_{k}^{2}=, \frac{\beta(\lambda_{k})}{\alpha(\lambda_{k})\sqrt{\lambda_{k}^{2}-1}}$and$\chi[-1,1](\lambda)$(3.6)

denotes the indicator function of [-1, 1].

(2)$d(n, m;\lambda)$$=$$\frac{\chi_{[-1,1]}(\lambda)d\lambda}{2\pi\sqrt{1-\lambda^{2}}}\{\psi^{+}(n;\lambda+i0)\overline{\psi^{+}(m,\cdot\lambda+i0)}$$+$$\psi^{+}(n;\lambda-i0)\overline{\psi^{+}(mj\lambda-i\mathrm{O})}$$+$$R(\lambda+i0)\psi^{+}(n;\lambda+i0)\overline{\prime\psi^{+}(m,\cdot\lambda-i\mathrm{O})}$$+$$R(\lambda-i0)\psi^{+}(n;\lambda-i0)\overline{\psi^{+}(m,\cdot\lambda+i\mathrm{O})}\}$$+$$\sum_{k=1}^{s}\psi^{+}(n;\lambda_{k})\psi^{+}(m.;\lambda_{k})c_{k}^{2}.\delta(\lambda-\lambda_{k})d\lambda$

(3.7)

24
For the proof see $[3],[6]$ .
We can rewrite (3.6),(3.7) by using the Fourier expnasions of$’’(n;$\psi^{+}(n;z)=\sum_{m\geq n}K(n, m)\zeta^{m}$$\ovalbox{\tt\small REJECT} \mathrm{z}$(3.8)$K(n, n)>0$$F(m)=F_{c}(m)+F_{p}(m)$ ,(3.9)$F_{c}(m)= \frac{1}{2\pi i}\int_{|\zeta|=1}R(z)\zeta^{n-1}’ d\zeta$(3.10)$F_{p}(m)=.

(3.11)\sum_{k=1}^{s}c_{k}^{2}\zeta_{k}^{m}$

We denote by
the operators defined by the kernel functions $\{F(n+$and.
is of Predholm type and of Hankel
type.
has abounded inverse.
Then (3.7) imply the following Gelfand-Levitan-Marchenko decomposition (abreviated by GLM decomposition)$K\wedge$$\hat{F},$$\hat{F}$$\{K(n, m)\}_{n,m=-\infty}^{\infty}$$m)\}_{n,m=-\infty}^{\infty}$$\hat{K}$

Proposition 3(1.1), (1.2) can be expressed in operator form as(3.12)(3.13)$1=\hat{K}(1+\hat{F})^{\mathrm{t}}\hat{K}$$A=\hat{K}A_{0}(1+\hat{F})^{t}\hat{K}=\hat{K}A_{0}\hat{K}^{-1}$

wheredenotes the transpose of . We denote bydiagonal matrix such that
, $a_{n}=0$ .
${}^{t}\hat{K}$$\hat{K}$$A_{0}$

the symmetric$t7\dot{?}-$$b_{n}= \frac{1}{2}$is positive definite so thathas the unique decomposition$1+\hat{F}$$A_{0}$$\hat{K}$is uniquely determined by (3.12).

(3.14)$A_{0}=A_{0,+}+A_{0,-}$

where
and
are upper triangular and lower triangular matrices respectively.
are unitary operators which shift the indices by
respec$A_{0,+}$$A_{0,-}$$2A_{0,\pm}$$\pm 1$

25
Now let us discuss how the $LR$ transform of can be expressed in termsof .
are bounded, there exists apositive number such that all 4
Since ,
$A_{0}(c)=A_{0}+c$ and $A(c)^{-1}$ , $A_{0}(c)^{-1}>0$ are positive
operators
definite.
We want to find the upper triangular -diagonal matrix $A_{+}(c)$ , withsuch that $\xi_{n}>0$ , and its
entriesandentriestranspose $A_{-}(c)={}^{t}A_{+}(c)$ , such that the following Gauss decompositon holds.
$A$$\hat{F}$

$A$

$c$

$A_{0}$$A(c)=\mathrm{A}(\mathrm{c})$$\mathrm{b}\mathrm{i}$$(n, n)\mathrm{t}\mathrm{h}$$(n, n+1)\mathrm{t}\mathrm{h}$$\xi_{n}$$\eta_{n}$(3.15)$A(c)=A_{-}(c)\cdot A_{+}\backslash (c)$i.e.,$\xi_{n}^{2}+\eta_{n-1}^{2}=a_{n}+c$,

The equations (3.16) have the unique solution such thatconvergent continued fraction$\xi_{0}^{2}=\frac{b_{0}^{2}|}{|a_{1}+c}-\frac{b_{1}^{2}|}{|a_{2}+c}-\cdots=-b_{0}’\frac{?l^{+}(1,-c)}{\psi^{+}(0,-c)}.\cdot$

because, if$z\not\in\sigma(A)$(3.16)$\xi_{n}\eta_{n}=b_{n}$$\xi_{0}^{2}$equals the(3.17), we have$\frac{b_{1}^{2}|}{z-a_{2}}-\cdots$$b_{0} \frac{\psi(1\cdot z)}{\psi(0\cdot z)},’=|z-a_{1} -| \underline{b_{0}^{2}|}$(3.18)We shall call the Gauss decomposition (3.15) thus obtained canonical.
The $LR$-transform is then defined as$Aarrow A’=A_{+}(c)\cdot A_{-}(c)=A_{+}(c)\cdot A\cdot A_{-}(c)^{-1}$$A’$

is also tri-diagonal.
We can now stat

(3.19)

26
Theorem 1Let the $GLM$ decompositon of $A’$ be(3.20)(3.21)$1=K’\cdot(1+\hat{F}’)\cdot{}^{t}\hat{K}’$$A’=K’\cdot A_{0}\cdot(1+\hat{F}’)\cdot{}^{t}\hat{K}’$then $A’$ is the $LR$ -transform of$A$

if and only if$\hat{F}’=\hat{F}\cdot A_{0,-}(c)\cdot A_{0,+}(c)^{-1}=A_{0,+}(c)\cdot\hat{F}\cdot A_{0,+}(c)^{-1}$(Remark that$\hat{F}\cdot A_{0,\pm}(c)=A_{0,\mp}(c)\cdot\hat{F}.$(3.22))

If we put$g( \zeta)=\frac{\sqrt{c+1}-\sqrt{c-1}}{2}\zeta+\frac{\sqrt{c+1}+\sqrt{c-1}}{2}$i.e.,$z+c=g(\zeta)g(\zeta^{-1})$

then (3.22) can be restated as$R’(z)=R(z)g(\zeta)^{-1}g(\zeta^{-1})$(3.23)

which is nothing else than dressing transformation in the sense of ZakhalovShabat. (This fact has been pointed out to the author by S.Kakei.)
Proof 1First we show that (3.22) implies (3.19). fi}vm
because of the uniqueness of Gauss decomposition, we have$\hat{K}’=A_{+}(c)\cdot\hat{K}\cdot g(2A_{0,+})$$(\mathit{3}.\mathit{2}\mathit{0}),(\mathit{3}.\mathit{2}\mathit{2})$and(3.24)

Hence, from (3.21)$A’=\hat{K}’\cdot A_{0}\cdot(1+\hat{F}’)\cdot {}^{t}K’=A_{+}(c)\hat{K}g(2A_{0,+})A_{0}g(2A_{0,-})^{-1}\hat{K}^{-1}A_{+}(c)^{-1}$$=A_{+}(c)\hat{K}A_{0}\hat{K}^{-1}A_{+}(c)^{-1}=A_{+}(c)\cdot A\cdot A_{+}(c)^{-1}$27(3.19) has thus been obtained.
Next we show the converse. We remark first that any bounded upper
is a holomorphic function of $2A_{0,+}$ .
triangular operator which commutes
As is seen from (3.12)and (3.19), there eists a holomorphic functionof ( $(|\zeta|<1)$ such that
$A_{0,+}$$\tilde{g}(\zeta)$(3.25)$\hat{K}’=A_{+}(c)\cdot\hat{K}\cdot\tilde{g}(2A_{0,+})$

Hence from (3.13), (3.20) and (3.21)$\tilde{g}(2A_{0,+})\tilde{g}(2A_{0,-})+\tilde{g}(2A_{0,+})^{2}\hat{F}’=A_{0}(c)^{-1}(1+\hat{F})$(3.26)

By uniquness of this matrix expression, we have$\tilde{g}(2A_{0,+})\tilde{g}(2A_{0,-})=A_{0}(c)^{-1}$(3.27)$\tilde{g}(2A_{0,+})^{2}\hat{F}’=A_{0}(c)^{-1}\hat{F}$(3.28)$\tilde{g}(2A_{0,+})=A_{0,+}\langle c)^{-1}$(3.29)$A_{0,+}(c)^{-2}\hat{F}’=A_{0}(c)^{-1}\hat{F}$(3.30)

which imply

andwhich are nothing else than (3.22).

4Periodic Toda lattice
Let

$A$

be aperiodic$\mathrm{t}\mathrm{r}\mathrm{i}$-diagonal matrix with period $N$ ,$a_{n+N}=a_{n}$(4.1), $b_{n+N}=b_{n}$. Let be the Floquet
We assume that it is positive definite on
multiplier andbe the $N\cross N$ matrix defined by$l^{2}(\mathrm{Z})$$A_{h}=(\tilde{a}_{n,m})_{n,m=0}^{N-1}$

$h$

28

$\tilde{a}_{n,m}=$$hb_{N-1}(n, m)=(N-1,0)$ ,$=$$h^{-1}b_{N-1}(n, m)=(0, N-1)$ ,$=$$a_{n,m}$otherwise

The determinant of $z-A_{h}$ can be written as$\det[z-A_{h}]=-b_{0}b_{1}\cdots b_{N-1}(h+h^{-1}-\Delta)$

where$\Delta$(4.2)

denotes the polynomial of degree $N$ such that$b_{0}b_{1}\cdots \mathrm{b}\mathrm{N}-\mathrm{i}\mathrm{A}=z^{N}-(a_{0}+a_{1}+\cdots+a_{N-1})z^{N-1}+\cdots$

The function$h$annhilating (4.2) is obtained by the equation$h= \frac{\Delta-\sqrt{\Delta^{2}-4}}{2}$(4.3)

which defines the hyperelliptic curve $X$ of genus $N-1$ .
Let , ,
be the roots of the equation $\Delta^{2}-4=0$ , such that$\lambda_{2N}$$\lambda_{1}$$\ldots$$0<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{2N-1}<\lambda_{2N}$$|h|=1$ i.e., $|\Delta|<4$ holds if and only ifA$\in[\lambda_{1}, \lambda_{2}]\cup[\lambda_{3}, \lambda_{4}]\cup\cdots\cup[\lambda_{2N-1}, \lambda_{2N}]$(4.4)

In other words, the spectra
are continuous and given by the bands (4.4).
, we have $|h|<1$ .
When ALet
be the Bloch solutions to (1.3) satisfying$\sigma(A)$$\not\in\sigma(A)$$\psi^{\pm}(n;z)$$\psi^{\pm}(n+N;z)=h^{\pm 1}\psi^{\pm}(n;z)$

(4.5)

29
which are obtained by solving the finite equations(4.6)$(z-A_{h})\tilde{\psi}=0$

Let $K^{\pm}(n;z)$ be the normalized Bloch solutions such that $K^{\pm}(0;z)=1$ .
We denote by $D(i, j)$ the subdeterminant corresponding to theentries $(i\leq n, m\leq j)$ of $z-A$ .
Then $K^{\pm}(n;z)$ can be expressed in terms of $D(i,j)$ , in particular$(n, m)\mathrm{t}\mathrm{h}$$K^{+}(1;z)=- \frac{(-1)^{N}hb_{1}\cdots b_{N-1}+b_{0}D(2,N-1)}{D(1,N-1)}$(4.7)$K^{-}(1;z)=- \frac{(-1)^{N}h^{-1}b_{1}\cdots b_{N-1}+b_{0}D(2,N-1)}{D(1,\mathit{1}\mathrm{V}-1)}$(4.8)

Proposition 4We havefor A

$K^{-}(n;\lambda+i0)=K^{+}(n;\lambda-i\mathrm{O})=\overline{K^{+}(n.\cdot\lambda+i\mathrm{O})}$$\in\sigma(A)$

(4.9)

Put
$d \rho_{+}(\lambda)=d\rho_{-}(\lambda)=\frac{1}{2\pi}\frac{|D(1,N-1)|}{|b_{0}b_{1}\cdots b_{N-1}|\sqrt{4-\Delta^{2}}}$

Then the spectral kernels of

$z4$

A

,

$\in\sigma(A)$can be expressed as(4.11)$d(n.m:\lambda)=\underline{9}\Re\{K^{+}(n:\lambda+i0)\overline{\mathrm{A}’+(’ nj\lambda+i0)}\}d\rho_{+}(\lambda)$

may put $D(1. N-1)$
the auxiliary spectra such that$\backslash \mathrm{t}^{r}\mathrm{e}$$= \prod_{k=1}^{\mathrm{V}-1}..(\approx-\mu_{k})\backslash \backslash \cdot \mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$$\mu_{1}$,$\mu_{2}$,$\lambda_{2}<l\iota_{1}<\lambda_{3}<\lambda_{4}<\cdots<\mu_{\mathrm{V}-1}.<\lambda_{2N-1}<\lambda_{2N}$$\backslash \backslash ^{r}\mathrm{e}$(4.10)

want to find the Gauss decompositon of$\wedge 4=A_{-}\cdot A_{+}$.

$\wedge 4$$A_{-}=.4_{\mathrm{A}}t$$\ldots$.

$\mu_{N-1}$

denote(4.12)

as in (3.15) (we put $c=0$ ).
$(’4.13)$30, $’/n+N$
such thathold.
We can find uniquely , rt. such that (3.16), (3.17) hold with \yen
$4_{n1N}\ovalbox{\tt\small REJECT}$\yen$E\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}.$$’/n$$C_{n}$(4.14)$\xi_{0}^{2}=-b_{0}K^{+}(1;0)$Remark that (3.17) is aperiodic continued fraction in this case.
The $LR$-transform is now defined by$A=A_{-}\cdot A_{+}arrow A’=A_{+}\cdot A_{-}=A_{+}\cdot A\cdot A_{+}^{-1}$(4.14)

The following Propsition 5is most fundamental.
Proposition 5Let
is the $LR$ -transform of

be the auxiliary spectra for $A’$ .
if and only if$\{\mu_{1}’, \mu_{2}’, \ldots, \mu_{N-1}’\}$

$A’$

$A$

$z. \frac{\prod_{k=1}^{N-1}(z-\mu_{k}’)}{\prod_{k=1}^{N-1}(z-\mu_{k})}=(\xi_{0}+\eta_{0}K^{+}(1;z))(\xi_{0}+\eta_{0}K^{-}(1;z))$The matrices $A’$ i.e.,(4.16).

$\xi_{n}$,$\eta_{n}$,$\mu_{1}’$,\ldots ,$\mu_{N-1}’$$I^{l}hen$(4.16)

can be uniquely obtained by solving

Proof 2The Bloch solutions for $A’$ are given by$K_{+}’(n;z)=.

\frac{\xi_{n}K^{+}(n,z)+\eta_{n}K^{+}(n.+1,z)}{\xi_{0}+\eta_{0}K_{+}(1,z)}.\cdot$(4.17)

We want to show first that
implies (4.15).
At $z=\infty$ , $K^{\pm}(n;z)$ are meromorphic and satisfy$(\mathit{4}\cdot \mathit{1}\mathit{6})$$K^{+}(n;z)=O(z^{-n})$ ,$T/iere$such that$K^{\prime+}(n;z)=O(z^{-n})$exists the unique upper triangular real matrix$K^{\prime+}(n;z)=, \sum_{n=n}^{\infty}\xi_{n,m}K^{+}(m;z)$$—=(\xi_{n,’ n})_{1\iota,m=-\infty}^{\infty}$(4.18)

31

From$(\mathit{4}\cdot \mathit{1}\mathit{6}),(\mathit{1}.\mathit{1})$and (1.2) we have the relations of operators$A/=—$.

$A \cdot\frac{\Pi_{k=1}^{N-1}(A-\mu_{k}’)}{\Pi_{k=1}^{N-1}(A-\mu_{k})}$. $t–=—-\cdot$

Moreover, there exists an upper triangular matr ix.

$t_{-}--$(4.19)$A\cdot----1$(4.20)$1=— \cdot.\frac{\Pi_{k=1}^{N-1}(A-\mu_{k}’)}{\Pi_{k=1}^{N-1}(A-\mu_{k})}$$\mathrm{Y}=(\eta_{n,m})_{n,m=-\infty}^{\infty}$$( \xi_{0}+\eta_{0}K^{+}(1;z))K^{+}(n;z)=\sum_{m=n}^{\infty}\eta_{n,m}K^{+}(m;z)$such that(4.21)

which is equivalent to the relations$\eta_{n,m}=2\int_{-\infty}^{\infty}\Re\{(\xi_{0}+\eta_{0}K^{+}(1;\lambda+i0))K^{+}(n;\lambda+i0)\overline{K^{+}(m,\cdot\lambda+i0)}\}d\rho_{+}(\lambda)$(4.22)

in into of (1.1) and (4.11). Therefore by substitution of
have

$A$

into $K^{+}(1;z)$ , we$\xi_{0}+\eta_{0}K^{+}(1;A)=\mathrm{Y}$(4.23)$\xi_{0}+\eta_{0}K^{-}(1;A)=\mathrm{Y}t$(4.24)In the same way,

From (4.16), these two equalities imply$A \frac{\prod_{k=1}^{N-1}(A-\mu_{k}’)}{\prod_{k=1}^{N-1}(A-\mu_{k})}=\mathrm{Y}\cdot {}^{t}Y={}^{t}\mathrm{Y}\cdot \mathrm{Y}$Since$\mathrm{Y}$and${}^{t}\mathrm{Y}$commute each other$(4’.25)$

32

A$\ovalbox{\tt\small REJECT}$

’Y .

’

$”\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}^{\ovalbox{\tt\small REJECT}_{1}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}^{A}}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$.Y$\ovalbox{\tt\small REJECT} {}^{t}Y\cdot t\ovalbox{\tt\small REJECT}$.

$\ovalbox{\tt\small REJECT}$.

$\mathrm{Y}$$\mathrm{I}\mathrm{L}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}(A \#\ovalbox{\tt\small REJECT})$

which is nothing else than the Gauss decomposition of , $i.e.$ ,$A$$A_{-}={}^{t}\mathrm{Y}\cdot \mathrm{t}---$,$A_{+}=---$.

$\mathrm{Y}$(4.26)

From (4.20)$A’=A_{+}\cdot \mathrm{Y}^{-1}\cdot A\cdot \mathrm{Y}\cdot A_{+}^{-1}=A_{+}\cdot A\cdot A_{+}^{-1}$which leads to (4.15).
Next we shoett that (4.15) implies (4.16).
Put$\psi’=A_{+}(K^{+})$(4.27)and normalize it such that $K^{;+}(0;z)=1$ as follows.

$K^{\prime+}(n; \approx)=,\frac{\psi’(n,z)}{\psi(0,\approx)}.\cdot$(4.28)

which gives (4.17) for $A’$ . Then there exists the unique upper tnangular ma. $—satisfying$. Hence,$tr\dot{\tau}x$$(\mathit{4}\cdot \mathit{1}\mathit{8})$$\{A_{+}(K^{+})\}(n;\sim\sim.)=\{(\xi_{0+7}\mathfrak{l}0^{K^{+}(1;z))_{-}^{-}}-(K^{+})\}(r\iota;\approx)=\mathrm{t}_{-}^{-}-.$$\mathrm{Y}(K^{+})\}(n;\sim)\sim$

In other words,$A+=—\cdot\}’$$l\mathit{4}s$a consequence(4.29)

33$\{A’\}_{n,m}=\{A_{+}\cdot A_{-}\}_{n,m}=\{_{-}^{-}-.

\mathrm{Y}\cdot {}^{t}\mathrm{Y}\cdot t---\}_{n,m}$$=2 \int_{-\infty}^{\infty}\Re\{(\xi_{0}+\eta_{0}K^{+}(1, \lambda+i0))(\xi_{0}+\eta_{0}K^{+}(1, \lambda-i0))$$K^{\prime+}$( , A $+i0$ )$n$$K^{\prime+}(m, \lambda-i0)\}d\rho_{+}(\lambda)$(4.30)On the other hand, by definition$\{A’\}_{n,m}=2\int_{\infty}^{\infty}\lambda\Re\{\lambda K^{\prime+}(n;\lambda+i0)K^{\prime+}(m;\lambda-i0)\}d\rho_{+}’(\lambda)$

Therefore by uniqueness of expression$\lambda d\rho_{+}’(\lambda)=(\xi_{0}+\eta_{0}K^{+}(1, \lambda+i0))$( $\xi_{0}+\eta_{0}K^{+}(1$ , A $-i0)$ )$d\rho_{+}(\lambda)$

Seeing that$d \rho_{+}’(\lambda)=\frac{\prod_{k=1}^{N-1}(\lambda-\mu_{k}’)}{\prod_{k=1}^{N-1}(\lambda-\mu_{k})}.d\rho_{+}(\lambda)$

we have (4.16).
The hyperelliptic curve defined by (4.3) has two sheets, physical andunphysical, which correspond to $|h|<1(>1)$ respectively, for.
Since $K^{\pm}(n;z)$ are meromorphic functions on $X$ , we can represent thefunctions $K^{\pm}(n;z)$ by using divisors in $X$ . Since $z=0$ ,are not branch$X$$X$
points of , there are two points inin each case, lying over $z=0$ , andin the physical sheet,in the unphysical sheet
respectively. $X$ has the canonical involution$\mathrm{X}$$\lambda\not\in\sigma(A)$$\infty$$z=\infty\langle 0\rangle,\langle\infty\rangle$$\langle 0^{*}\rangle,\langle\infty^{*}\rangle$$\iota$Obviouslyof adivisor

and$D$. Then$\iota(\{0\rangle)=\langle 0^{*}\rangle$$\iota(D)$: $harrow h^{-1}$ .

$\iota(\langle\infty\rangle)=\langle\infty^{*}\rangle$(4.31). We denote by$D^{*}$

the conjugate

Lemma 1Fix $n\geq 0$ . $K^{+}(n;z)$ has simple poles at the physical points in$X$ , lying
over $z=\mu_{1}$ , , ,which do not depend on . We denote thecorresponding positive divisor of degree $N-1$ by Do. It has also a pole of$\mu_{2}$$\ldots$$\mu_{N-1}$

$n$

34
order n at \langle --,\rangle . Similarly it has simple zeros at the unphysical points lying)(its divisor of degree N-1 is denoted byover Z j’r\rangle
a zero of order n at \langle - \rangle .
In other words, in terms of divisors,$D_{n}$$\ovalbox{\tt\small REJECT}$$/’ \mathit{2}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\rangle$$art_{\ovalbox{\tt\small REJECT}}d$$j\ovalbox{\tt\small REJECT} y-+$(4.32)(4.33)$(K^{+}(n;z))=n\langle\infty\rangle-n\langle\infty^{*}\rangle-D_{0}+D_{n}$$(K^{-}(n;z))=n\langle\infty^{*}\rangle-n\langle\infty\rangle-D_{0}^{*}+D_{n}^{*}$

Furthermore$( \prod_{k=1}^{N-1}(z-\mu_{k}))=-(N-1)\{\langle\infty\rangle+\langle\infty^{*}\rangle\}+D_{0}+D_{0}^{*}$(4.34)$(h)=N(\langle\infty\rangle-\langle\infty^{*}\rangle)$(4.35)

As for the zeros and poles of $\xi_{0}+\eta_{0}K^{+}(1;z)$ , we have
Theorem 2There exist a positive divisor of degree $N-1$ ,, such thatjugate$D_{0}’$and its con-

$D_{0}^{\prime 1}$(4.36)(4.37)$(\xi_{0}+\eta_{0}K^{+}(1;z))=\langle 0\rangle-\langle\infty^{*}\rangle-D_{0}+D_{0}’$$(\xi_{0}+\eta_{0}K^{-}(1;z))=\langle 0^{*}\rangle-(\mathrm{o}\mathrm{o})-D_{0}^{*}+D_{0}’*$Hence, there eists a positive divisor of degree $N-1$ ,$D_{1}’$$such$

that(4.38)$(K^{\prime+}(1;z))=\langle\infty\rangle-\langle\infty^{*}\rangle-D_{0}’+D_{1}’$The set of divisor classes of degree $N-1$ in $X$ makes the Jacobi variety
of $X$ denoted by $Jac(X)$ . As is seen from (4.36), we have the equality as apoint of $Jac(X)$(4.39)$D_{0}’-D_{0}\equiv-\langle 0\rangle+\langle\infty^{*}\rangle$

The new -diagonal operator $A’$ has the same spectra as and therefore
we can take the $LR$-transform of $A’$ again. By repeating this procedure, we
get asequence of -diagonal operator$A$$\mathrm{t}\mathrm{r}\mathrm{i}$$\mathrm{t}\mathrm{r}\mathrm{i}$35
A $arrow A’arrow A’arrow\cdots$(4.40)

and asequence of corresponding divisor classes$D_{0}arrow D_{0}’arrow D_{0}^{\prime/}arrow\cdots$(4.41)$D_{0}’-D_{0}\equiv D_{0}’-D_{0}’\equiv\cdots\equiv-\langle 0\rangle+\langle\infty^{*}\rangle$(4.42)

such thatAs aconclusion,
Theorem 3The sequence of $LR$ -tranforms(4.40) is realized in $Jac(X)$ , byby the constant divisor $class-\langle 0\rangle+$
the discrete paralell displacement of, starting fromsuch that$\mathrm{p}_{m}$$\mathrm{P}\mathrm{o}=D_{0}$$\langle\infty^{*}\rangle$$\mathrm{p}_{m}=\mathrm{P}\mathrm{o}+m\{-\langle 0\rangle+\langle\infty^{*}\rangle\}$,$m=0,1,2,3$ ,$\ldots$.

(4.43)

Corollary 1The sequence of $LR$ -transforms is periodic with per $.odM>0$
if and only if(4.44)$M\{-\langle 0\rangle+\langle\infty^{*}\rangle\}\equiv 0$

Remark 1When is finite or semi-infinite, the sequence (4.40) never become periodic. In fact, in a finite case, tends to a diagonal matr , so thatthe eigenvalues of are approximated by these procedure([25],[26],[27]). I do
not know how they behave, when is semi-finite.
$A$

$A$

$ix$

$A$

$A$

Remark 2 $f(z)$ is a polynomial of degree , it is possible to extend (4.16)
to a more general transform (1.6). In this situation (4.16) must be replacedby the equation$r$$f(z) \frac{\Pi_{k^{\wedge}=1}^{N-1}(z-\mu_{k}’)}{\Pi_{k=1}^{N-1}(z-\mu_{k})}=(\xi_{0}+.\sum_{k=1}^{r}\eta_{0,k}K^{+}(k;z))(\xi_{0}+\sum_{k=1}^{f}\eta_{0,k}K^{-}(k;z))$36
Since $f(A)$ is no more $tri$ -diagonal, we cannot find .-diagonal matrices
satisfying (1.5).
Suppose that $f(A)$ is positive definite and $m,ultiple$ -diagonal of width $2m+$. Then $f(A)$ is a .-diagonal matrix in block form, consisting of matrices$An,n$ , $(A_{n,n}={}^{t}A_{n,n}>0)A_{n,n+1}$ , $A_{n+1,n}={}^{t}A_{n,n+1}$
of size $m+1$ . One canfind an upper block -triangular matrix consisting of triangular matrices
andof size $m+1$ such that$tr$

$B_{\pm}$

$tr$

$1$

$bi$

$B_{+}$$B_{n,n+1}$$B_{n,n}$$A_{n,n}.={}^{t}B_{n,n}\cdot B_{n,n}+{}^{t}B_{n-1,n}\cdot B_{n-\mathrm{I},n}$If we put

$Z_{n}={}^{t}B_{n,n}\cdot B_{n,n}$, $A_{n.,n+1}={}^{t}B_{n,n}\cdot B_{n,n+1}$, then we have the reccurence relations$Z_{n}=A_{n,n+1}\cdot(A_{n+1,n+1}-Z_{n+1})^{-1}\cdot {}^{t}A_{n,n+1}$

width give the matrix version of the convergent continuedfraction (3.17) suchthat
$Z_{n}\leq A_{n,n+1}\cdot A_{n+1,n+1}^{-1}\cdot {}^{t}A_{n+1,n}$$B_{n,n}$can be solved uniquely from$Z$

such that all the diagonal elements are

positive.

In the next section, in case of N $=2$ , we shall give explicit computationin terms of the sigma functions on the elliptic curve X.

5Case of period N $=2$
It is sufficient to give
We put$\{a_{0}, a_{1}, b_{0}, b_{1}\}$$W(z)=b_{0}^{2}b_{1}^{2}(\Delta^{2}-4)$

to define the operator A.

, then$W(z)=(z-\lambda_{1})(z-\lambda_{2})(z-\lambda_{3})(z-\lambda_{4})$,$0<\lambda_{1}<\lambda_{2}<\lambda_{3}<\lambda_{4}$

(5.1)

,

$\lambda_{2}<a_{1}<\lambda_{3}$(5.2)

Moreover$d \rho_{\pm}=\frac{1}{4\pi}\frac{|\lambda-a_{1}|}{\sqrt{|W(\lambda)|}}$$K^{+}(1;z)= \frac{b_{0}+b_{1}h}{z-a_{1}}$,$K^{-}(1;z)= \frac{b_{0}+b_{1}h^{-1}}{z-a_{1}}$

,

(5.3)

37
.

(4. 16) reduces to$\cdot$(5.’$z \frac{z-a_{1}’}{z-a_{1}}=(\xi_{0}+\eta_{0}K^{+}(1;z))(\xi_{0}+\eta_{0}K^{-}(1;z))$Put$u= \int_{\lambda_{4}}^{z}\frac{dz}{\sqrt{W(z)}}$,$v= \int_{\lambda_{4}}^{\infty}\frac{dz}{\sqrt{\mathrm{L}V(z)}}>0$,$v-c= \int_{\lambda_{4}}^{a_{1}}\frac{dz}{\sqrt{W(z)}}$, $v>\Re c>0$ , $\Im c<0$

$w= \int_{\lambda_{4}}^{0}\frac{dz}{\sqrt{W(z)}}>0$$\omega_{1}=\int_{\lambda_{2}}^{\lambda_{3}}\frac{dz}{\sqrt{W(z)}}>0,\omega_{2}=i\int_{\lambda_{3}}^{\lambda_{4}}\frac{dz}{\sqrt{|W(z)|}}\in i\mathrm{R}_{>0}$then,$2\mathrm{w}\mathrm{a}$)$2\omega_{2}$are double periods, and$\langle 0\rangle$,$\langle 0^{*}\rangle$,$\langle\infty\rangle$$u=w$ , $u=-w$ , $u=v$ , $u=-v$

respectively. Furthermore,$4v=2\omega_{1}\equiv 0$$\mathrm{i}.\mathrm{e}.$,$D_{2}-D_{0}\sim 0$$\sigma(u)$has the zero $u=0$ , and quasi-periodic$\sigma(u+2\omega_{1})=-e^{2(\eta_{1}u+\omega_{1})}\sigma(u)$$\sigma(u+2\omega_{2})=-e^{2(\eta\circ u+\iota v_{2})}\sim\sigma(u)$(where$\eta_{1}$,$\eta_{2}$denote constants). We have,$\langle\infty^{*}\rangle$

correspond to38$z=- \frac{\sigma(u+w)\sigma(u-w)\sigma(2v)}{\sigma(u-v)\sigma(u+v)\sigma(v+w)\sigma(v-w)}$$h=C_{1} \frac{\sigma^{2}(u-v)}{\sigma^{2}(u+v)}$$K^{+}(1;z)=C_{2} \frac{\sigma(u-v)\sigma(u+v+c)}{\sigma(u+v)\sigma(u-v+c)}$,$K^{+}( \prime 1;z)=C_{3}\frac{\sigma(u-v)\sigma(u+v+d)}{\sigma(u+v)\sigma(u-\mathrm{c}+d)}$

If we put$c’-c=v+w= \int_{0^{*}}^{\infty}\frac{dz}{\sqrt{W(z)}}$then the $LR$-transform represents the paralell displacement on the 1dimensional complex trorus$\mathrm{C}/(\mathrm{Z}2\omega_{1}+\mathrm{Z}2\mathrm{u}2)$$carrow c+v+warrow c+2(v+w)arrow\cdots$

In order that it is periodic, there exists apositive integer $M$ such that$M(v+w)\equiv 0$$(2\omega_{1},2\omega_{2})$6Multi-Index Hankel Matrices and Orthogonal Polynomials in Multi-Variables
In the next three sections we shall make amulti-dimensional extension of
LR-transforms developed in the previous sections . Multi-dimensional
LR-transforms are related with eigenfunction expansions for commuting
self-adjoint operators.
We restrict ourselves to orthogonal polynomials case. The problem offinding -transforms reduces to obtaining the connection formula between
two systems of orthogonal polynomials. Our main result in this section is$\mathrm{L}\mathrm{R}$39
Theorem 4. In the course of proof, we shall give aformula for the connection
matrix which is alower triangular matrix, in terms of determinants of theassociated muti-dimensional Hankel matrix.
Let $d\rho=d\rho(x)$ , $x=(x_{1}, \cdots, x_{n})$ be aRadon measure on the dimensional Euclidean space
whose support is abounded closed set 7). We
assume that all multi-index moments$n$$\mathrm{R}^{n}$(6.1)$c_{i_{1},\cdots,i_{n}}= \int_{\mathrm{R}^{n}}x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}d\rho(x)$are finite.
Let$H_{\rho}$be the Hilbert space completed by the inner product /,$($$g)_{\rho}$(6.2)$(f, g)_{\rho}= \int_{\mathrm{R}^{n}}f(x)g(x)d\rho(x)$for real continuous functions $f(x)$ , $g(x)$ on.
For two sequences of indices $I=$
and $J=(j_{1}, \cdots,j_{n})$ , wedefine the sum $I+J$ by the sequence of indices $(i_{1}+j_{1}, \cdots, i_{n}+j_{n})$ .
We define the lexicographic ordering
for the set of multi-indices asfollows.
is greater than
if and only if there exists anumber$r(1\leq r\leq n)$ such that,
, $i_{r-1}=j_{f-1}$ ,. In this case, we also
say the monomial
is greater than the monomial.
Thus we have the sequence of monomials in increasing order$\mathrm{R}^{n}$$(i_{1}, \cdots, i_{n})$$\mathcal{O}$$(i_{1}, \cdots, i_{n})$$(j_{1}, \cdots,j_{n})$$i_{1}=j_{1}$$\cdots$$x_{1}^{i_{1}}\cdots x_{n}^{i_{n}}$$i_{r}>j_{\tau}$$\dot{d}_{1}^{1}\cdots x_{n}^{j_{n}}$$1<x_{1}<\cdots<x_{n}<x_{1}^{2}<x_{1}x_{2}<\cdots<x_{7l}^{2}<x_{1}^{3}<\cdots$

Let $N$ be the unique bijective mapping from the set of positive integers
onto the set of multi-indices such that $N(l_{1})<N(l_{2})$ for two positive integers.
We have $N(1)=(0, \cdots, 0)$ , $N(2)=(1, \cdots, 0)$ , $N(3)=(0,1,0, \cdots, 0)$ ,, $N(n+1)=(0, \cdots, 0,1)$ , and $N(n+2)=(2,0, \cdots, 0)$ etc.
We assume that $d\rho(x)$ is non-degenerate in the sense that$l_{1}<l_{2}$$\ldots$$\int_{\mathrm{R}^{n}}f(x)^{2}d\rho(x)>0$

for any polynomial $f(x)$ which is not identically zero on 7).

40
Let C be the generalized Hankel matrix with the $N(7)$ ,entriesfor \yen It is apositive definite matrix ? so that all
the determinants$N(\mathrm{r}\mathrm{r}\mathrm{z})\mathrm{t}\mathrm{h}$$\mathrm{c}_{N(\mathrm{J})+N(m)}$0$\mathrm{j}^{\ovalbox{\tt\small REJECT}})^{\ovalbox{\tt\small REJECT}})$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$D_{N(\mathrm{r})}=\det((c_{N(l)+N(m)})_{l,m=1}^{r})>0$

for $(0\leq r<\infty)$ . Here we put $D_{N(0)}=1$ .
Gram-Schmit orthonormalization with respect to the lexicographic ordering gives the orthonormalized polynomials
such that$\{p_{i_{1},\cdots,i_{n}}\}_{i_{1},\cdots,:_{n}\geq 0}$$p_{i_{1},\cdots,i_{n}}=\xi_{i_{1},\cdots,i_{n}}x_{1}^{i_{1}}\cdots$ $x_{n}^{i_{n}}+$(tower order terms)

(6.3)

where
denote normalizing positive constants. Hence we have the orthonormality$\xi_{i_{1},\cdots,i_{n}}$(6.4)$(p:_{1},\cdots,i_{n},p_{j_{1\prime\prime}j_{n}}\ldots)_{\rho}=\delta_{i_{1},j_{1}}\cdots\delta:_{1},,j_{n}$Let $N(l)$ denote the multi-index. Then the polynomials
nant$(i_{1}, \cdots, i_{n})$$x^{N(l)}=x_{1^{1}}^{\dot{1}}\ldots$$x_{n}^{i_{n}}$$c_{N(2)}c_{N(1)}$. We denote the monomial
defined by the determi-$\tilde{p}_{i_{1},\cdots,i_{\hslash}}(x)$$c_{N(1)+N(2)}c_{N(2)}$$c_{N(2)+N(l)}c_{N(l)}$$\tilde{p}_{i_{1\prime\prime}i_{n}}\ldots(x)=\frac{1}{D_{N(l-1)}}$$\cdot$$c_{N}x^{N}(\begin{array}{l}l-1()\end{array})$$c_{N(l-1)+N(2)}x^{N(2)}$

.

(6.5)
$\cdot$. .

$c_{N(l-1)+N(l)}x^{N(l)}$$\cdot$are monic orthogonal polynomials such that the following equations hold.
$(\tilde{p}_{i_{1},\cdots,i_{n}}(x),\tilde{p}_{j_{1\prime\prime}j_{n}}\ldots(x))=0$

for$( \tilde{p}_{i_{1},\cdots,i_{\iota}},(x),\tilde{p}_{i_{1},\cdots,i_{\mathfrak{n}}}(x))=\frac{D_{N(l)}}{D_{N(l-1)}}$$(i_{1}, \cdots, i_{n})\neq(j_{1}, \cdots,j_{n})$(6.6)

for $N(l)=(i_{1}, \cdots, i_{n})$(6.7)

so that we have the orthonormalized polynomials$p_{N(l)}(x)=$

(6.3)

41
(The above computation can be done in the same way as in [31].)
We have$p_{N(l)}(x)=$Let$A_{1}$,$A_{2}$,$\cdots$,$\cdots x_{n}^{i_{n}}+$$A_{n}$( lower order terms)

be the bounded linear operators on$A_{j}\varphi(x)=x_{j}\varphi(x)$$7\{_{\rho}$defined by(6.9)$\varphi(x)\in H_{\rho}$

They can be expressible in matrix form
$x^{N(l)}l=1,2,3$ ,in terms of the basis$a_{N(l),N(m)}^{(j)}$$\cdots$$x_{j}p_{N(l)}(x)=$$\sum$$m\geq 1$$(1 \leq j\leq n)$$a_{N(l),N(m)}^{(j)}pN(m)(\prime x)$(6.10)(finite sum)

We have(6.11)$a_{N(l),N(m)}^{(j)}=(x_{j}p_{N(l)}(x),p_{N(m})(x))_{\rho}$$A_{j}$

are se -adjoint bounded operators and commute each other.
Letdenote the Hilbert space consis ting of real sequences $u=$
$1\mathrm{f}$$L^{2}(\mathrm{Z}_{\geq 0}^{n})$$(u_{i_{1},\cdots,i_{n}})_{i_{1},\cdots,i_{n}\geq 0}$

with the inner product$\mathit{8}l$$(u, v)= \sum_{i_{1},\cdots,i_{\tau\iota}\geq 0}u_{i_{1,\prime}i_{n}}\ldots v_{i_{1},\cdots,i_{n}}$,$v\in L^{2}(\mathrm{Z}_{\geq 0}^{n})$The correspondence from the set of real sequences
continuous functions$(u_{i_{1},i_{2},\cdots,i_{n}})_{i_{1},\cdots,i_{n}\geq 0}$

to$\varphi(x)$(6.12)$(u_{i_{1},i_{2},\cdots,i_{n}})_{i_{1},\cdots,i_{n}\geq 0} arrow\varphi(x)=\sum_{i_{1},\cdots,i_{n}\geq 0}?\iota_{i_{1},i_{2},\cdots,i_{n}}p_{i_{1},\cdots,i_{n}}(x)$give rise to the isomorphism between the space,
Consider the shifts for the sequences$L^{2}(\mathrm{Z}_{\geq 0}^{n})$$\tau_{\nu}$$\tau_{\nu}^{\pm}:$$.i_{1}\geq$$\cdots$,and$i_{n}\geq 0$$?\{_{\rho}$.

as(6.10)$(i_{1}, \cdots, i_{n})arrow(i_{1}, \cdots, i_{\nu}\pm 1, \cdots i_{n})$the number
For $N(l)=(i_{1}, \cdots, i_{n})$ , we denote bysuch thatby abuse of notation (Remark that
does not exist when$i_{\nu}=0$ . )From the relations (6.5),(6.8) and (6.10) the following Proposition holds$l^{\pm}$$\tau_{\nu}^{\pm}l$$N(l^{\pm})=\tau_{\nu}^{\pm}N(l)$$l^{-}$

42
Proposition 6Assume 1
aselements$\ovalbox{\tt\small REJECT}$m$\ovalbox{\tt\small REJECT}$

1. We can represent explicitly the matrix$\mathrm{C}\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}})$$a/\mathrm{v}\ovalbox{\tt\small REJECT}_{\mathrm{J})_{\ovalbox{\tt\small REJECT}}N(m)}\ovalbox{\tt\small REJECT}$$a_{N(l),N(m)}^{(j)}=$$\sum$$(-1)^{l+m+\mathrm{r}_{\frac{1}{\sqrt{D_{N(l)}D_{N(l-1)}D_{N(m)}D_{N(1n-1)}}}}}$$m\leq r\leq l$.

$|\begin{array}{llll}c_{N(1)} c_{N(2)} \cdots c_{N(m)}c_{N(2)} c_{N(1)+N(2)} \cdots c_{N(2)+N(m)}\cdots \cdots \cdots \cdots c_{N(m-1)} c_{N(m-1)+N(2)} \cdots c_{N(m-1)+N(}n)c_{N(r)} c_{N(\tau)+N(2)} \cdots c_{N(\mathrm{r})+N(m)}\end{array}|$.

$(6.14)$$|\rangle.\cdot c_{\tau_{j}^{-}N(r)}.\cdot c_{N(2)}c_{N(1)}c_{N(l)}.\cdot\cdot.\cdot..\cdot\cdot.\cdot$$\cdot..\cdot c_{\tau_{j}^{-}N(\mathrm{r})+N(2)}..c_{N(1)+N(2)}c_{N(l)+N(2)}.\cdot.\cdot c_{N(2)}.\cdot.\cdot\cdot..\cdot\cdot..\cdot\cdot..\cdot.\cdot..\cdot\cdot..\cdot.\cdot.\cdot.\cdot...\cdot.\cdot c_{\tau_{j}^{-}N(\tau)+N(l-1)}.\cdot\cdot.\cdot..\cdot.\cdot,\cdot.\cdot\cdot..\cdot.\cdot\cdot..\cdot,\cdot.\cdot\langle c_{N(2)+N(l-1)}c_{N(l)+N(l-1)}c_{N(l-1)}$.
denotes the deletion of aline)
which is non-negative on V.
Let $f(x)$ be acontinuous function onsame support as V.
withon
Consider the new density(The symbol$\rangle\cdots\langle$$\mathrm{R}^{n}$$d\rho’(x)$$\mathrm{R}^{n}$$\mathrm{t}\cdot \mathrm{h}\mathrm{e}$(6.15)$d\rho’(x)=f(x)d\rho(x)$Then we can define the multiplication operators$A_{j}’$:$\varphi(x)arrow x_{j}\varphi(x)$on the new Hilbert space(6.16)$(1 \leq j\leq n)$.
with the inner productand be the correLet
be the moments of the density,
entries $c_{N(l)+N(m)}’$ .
sponding generalized Hankel matrix with the $N(l)$ ,, which is positive
Then $f(A_{1}, \cdots, A_{n})$ is aself-adjoint operator on
definite, because$(\cdots, \cdots)_{\rho’}$$\prime H_{p’}=L^{2}(\mathrm{R}^{n};d\rho’)$$d\rho’$$(d_{i_{1},\cdots,i_{n}})_{i_{1}i_{n}\geq 0}.’\ldots$$\mathrm{C}’$$N(m)\mathrm{t}\mathrm{h}$$H\rho$$(f(A_{1}, \cdots, A_{n})\varphi(x), \varphi(x))_{\rho}=\int_{D}\varphi(x)^{2}f(x)d\rho(x)>0$

for acontinuous function$\varphi(x)$which does not vanish identically in

$D$

.

43
Let
be the Gram-Schmidt orthonormalization according to the lexicographic ordering .
are defined similarly to (1.5), replacingby$(p_{i_{1},\cdots,i_{n}}’(x))_{i_{1},\cdots,i_{n}\geq 0}$$O$$(\tilde{p}_{i_{1},\cdots,i_{n}}’(x))_{i_{1},\cdots,i_{n}\geq 0}$$c_{i_{1}}’,\cdot.\mathrm{T}^{\cdot}\mathrm{h}\mathrm{e}$$c_{i_{1},\cdots,i_{n}}$operator $f(A_{1}, \cdots, A_{n})$ can be represented by the matrix with theelements $(f(A_{1}, . \cdots, A_{n})p_{N(l)}(x),p_{N}(m)(x))_{\rho}$.
We are interested in the connection relations between the two set of orthogonal polynomials
andcan be represented as alinear combination of$N(l)$ ,$N(m)\mathrm{t}\mathrm{h}$$(p_{i_{1},\cdots,i_{n}})_{i_{1},\cdots,i_{n}}$$(p_{i_{1},\cdots,i_{n}}’)_{i_{1},i_{n}}.\cdots,\cdot$$p_{j_{1},\cdots,j_{n}}’$$p_{i_{1},\cdots,i_{n}}$$p_{N(l)}(x)= \sum_{\backslash }R_{N(l)/N(m)I^{J_{N(m)}’}}(x)1_{-}’m\leq l$(6.17)

We put further $R_{N(l)/N(m)}$ to be 0for $l<m$ , so that $R=(R_{N(l)/N(m)})_{l,m\geq 0}$
defines an invertible lower triangular matrix with respect to the lexicogrphic
ordering. In particular the diagonal elements are expressed as$>0$$R_{N(l)/N(l)}=$As for the relations between$\tilde{p}_{i_{1},\cdots,i_{n}}$and$\tilde{p}_{i_{1},\cdots,i_{n}}’$(6.18), we have similarly,$\tilde{p}_{N(l)}=\sum_{1\leq m\leq l}\tilde{R}_{N(l)/N(m)}\tilde{p}_{N(n)}’$(6.19)

for an invertible lower triangular matrix. Remarkthat $\tilde{R}_{N(l)/N(l)}=1$ . In view of (6.8),(6.17) and (6.19), the following identities
hold.
$\tilde{R}=(\tilde{R}_{N(l)/N(m)})_{1\leq l,m<\infty}$$R_{N(l),N(m)}=$(6.20)

Theorem 4As a matrix expression, we have$f(A_{1}, \cdots, A_{n})=R\cdot {}^{t}R$(6.21)

44
The matrix R is uniquely deter mined by (L21).
For every j, we have the following LR-transforms(6.22)$A_{j}’=R^{-1}\cdot A_{j}\cdot R$

In parti cular,$f(A_{1}’, \cdots, A_{n}’)=R^{-1}\cdot f(A_{1}, \cdots, A_{n})\cdot R={}^{t}R\cdot R$

which is just the interchange of$R^{-1}$$R$and${}^{t}R$.

$R$

is an invertible matrix so thatis well-defined.

For
expression$u=(u_{i_{1},\cdots,i_{n}}):_{1},\cdots,i_{n}\in L^{2}(\mathrm{Z}_{\geq 0}^{n})$, (6.10) and (6.12) give the matrix(6.23)$(A_{j}u):_{1,\prime} \cdots:_{n}=\cdots\sum_{j_{1,\prime}j_{\hslash}\geq 0}a_{(1,\prime i_{n}),(j_{1},\cdots,j_{n})}^{(j)}.u_{j_{1\prime\prime}j_{n}}|\cdots\ldots$.
is is0be the Hilbert space spanned by the sequencesLet
such thatinmorphic to the space of sequences$(f(A_{1}, \cdots, A_{n})^{-1}v, v)<\infty$ .Then the inverse
is well-define as bounded.
to
operator fromcan be expressed by using the following
The matrix elements,, ,for different positive integers ,system of determinantsfrom each other.
${}^{t}Ru$$H_{0}$$v=(v_{i_{1},\cdots,i_{\hslash}})_{i_{1},\cdots,:_{\hslash}}$$L^{2}(\mathrm{Z}_{\geq 0}^{n})$$R^{-1}$$H_{0}$$H_{0}$$\mathrm{d}$$L^{2}(\mathrm{Z}_{\geq 0\grave{J}}^{n}$$\tilde{R}_{N(l),N(m)}$$l_{1}$$\psi_{l_{1},\cdots,l_{\mathrm{r}}}$$\psi_{l_{1}}=c_{N(l_{1})}$,$\cdots$$\cdots$.

, 0).)In the same way we define the determinants
moment$d_{N(l)}$$l_{r}$$\psi_{l_{1},l_{2}}=|\begin{array}{ll}c_{N(l_{1})} \mathrm{c}_{N(l_{2})}c_{N(\mathit{2})+N(l_{1})} c_{N(2)+N(l_{2})}\end{array}|$$\psi_{l_{1},l_{2\prime\prime}l_{f}}\ldots=|c_{N(2)+N(l_{1})}\ldots\ldots\ldots\ldots c_{N(2)+N(l_{2})}\ldots\ldots..\cdot\ldots.\cdot.\cdot.\cdot\ldots c_{N(2)+N(l_{r})}c_{N(r)+N(l_{1})}c_{N(\mathrm{r})+N(l_{2})}\cdots c_{N(r)+N(l_{f})}c_{N(l_{1})}c_{N(l_{2})}\cdots..c_{N(l_{\mathrm{r}})}\ldots\ldots$(Remark that $N(1)=(0,0$,$\cdots$$.\psi_{l_{1},\cdots,l_{t}}’$associated with the45$\psi_{l_{1}}’=c_{N(l_{1})}’$,$\psi_{l_{1},l_{2}}’=|\begin{array}{ll}d_{N(l_{1})} c_{N(l_{2})}’c_{N(2)+N(l_{1})}’ c_{N(2)+N(l_{2})}’\end{array}|$$\psi_{l_{1},l_{2},\cdots,l_{r}}’=|\begin{array}{llll}d_{N(l_{1})} c_{N(l_{2})}’ \cdots c_{N(l_{T})}’c_{N(2)+N(l_{1})}’\prime c_{N(2)+N(l_{2})}’ \cdots d_{N(2)+N(l_{\Gamma})}\cdots \cdots \cdots \cdots c_{N(\mathrm{r})+N(l_{1})}’ c_{N(r)+N(l_{2})}’ \cdots c_{N(r)+N(l_{\prime})}’\end{array}|$

Then we have
Proposition 7$\tilde{R}_{N(l),N(m)}$1$=$$D_{N(m)}’\cdots D_{N(l-1)}’D_{N(l-1)}$$\sum\epsilon\psi_{1,2,\cdots,m-1,\alpha_{n\iota,,n}}’\psi_{1,2,\cdots,m-1,\alpha_{m+1.m},\alpha_{m+1.m+1}}’$$\ldots$,,
wheremove \^o er the set
that the following identities hold as sets$\alpha_{m,m}$$\alpha_{m+1,m}$(6.24)$\psi_{1,2,\cdots,m-1,\alpha_{l-1.m},\cdots,\alpha_{l-1,l-1}}’\psi_{1,2,\cdots,m-1,\alpha_{l.m},\cdots,\alpha_{l,l-1}}$$\cdots$

finite sequences of integers such$\{\alpha_{m,n},, \alpha_{m+1,m}\}$$=$$\{m, m+1\}$ ,$\{\alpha_{m+1,m+1}, \alpha_{m+2,m}, \alpha_{m+2,m+1}\}$$=$$\{m, m+1, m+2\}$ ,$\{\alpha_{l-2,l-2}, \alpha_{l-1,m}, \cdots, \alpha_{l-1,l-2}\}$$=$$\{m, m+1.

$\{\alpha_{l-1,l-1}, \alpha_{l,m}, \cdots, \alpha_{l,l-1}\}$$=$$\{n\tau, m+1, \cdots, l\}$\cdots, l-1\}$ ,and that$\alpha_{m+2,m}<\alpha_{m+^{9}\sim.m+1}$,$\alpha_{l-1,m}<\alpha_{l-1,m+1}<\cdots<\alpha_{l-1,l-2}$,$\alpha_{l,m}<\alpha_{l,m+1}<\cdots<\alpha_{l,l-1}$$\epsilon$

denotes the suitably chosen $sign\pm depending$ on the choices of$\alpha’ s$.

46This Proposition follows by solving (6.19) term by term in view 0f(6.5).
For example,,$\tilde{R}_{N(l),N(l)}$

$=$

$1$

$\tilde{R}_{N(l),N(l-1)}$$=$$D_{N(l-1)}’D_{N(l-1)}1$$\tilde{R}_{N(l),N(l-2)}$$=$$(\psi_{1,2,\cdots,l-2,l}’\psi_{1,2,l-2,l1},\cdot--\psi_{1,2,\cdots,l-2,l-1}’\psi_{1,2,\cdots,l-2,l})D_{N(l-1)}’D_{N(l-2)}’D_{N(-}1l1)$($\psi_{1,2,\cdots l-2}’\psi_{1,2,\cdots l-1}’\psi_{1,2,\cdots l-3,l-1,l}$\prime\prime\prime$\psi_{1,2,\cdots l-3,l-1}’\cdot\psi_{1,2,\cdots l-2,l}’\psi_{1,2,\cdots,l-1}$\prime\prime$+$$\psi_{1,2,\cdots,l-2}’\psi_{1,2,\cdots,l-3,l-1,l}’\psi_{1,2,\cdots,l-1}$-$\psi_{1,2,\cdots l-3,l-1}’\psi_{1,2,\cdots l-1}’\psi_{1,2,\cdots,l-2,l}$)\prime\primeand so on.
Example. (Appell’s Polynomials) Suppose the density(6.25)$d\rho(x)=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}(1-x_{1}-\cdots-x_{n})^{\alpha_{\iota+1}}\cdot dx_{1}\wedge\cdots\wedge dx_{n}$be defined on the simplex 7) :$x_{1}\geq 0$,$\cdots$, $x_{n}\geq 0$ , $x_{1}+\cdots+x_{n}\leq 1$ . We have$c_{i_{1\prime\prime}i_{n}} \ldots=\frac{\Gamma(\alpha_{1}+i_{1}+1)\cdots\Gamma(\alpha_{n}+i_{n}+1)\Gamma(\alpha_{n+1}+1)}{\Gamma(\alpha_{1}+\cdots+\alpha_{n+1}+i_{1}+\cdots+i_{n}+n+1)}$

The ratio
are rational functions ofrational function of ,and the monic polynomial,. Whence every element$\mathrm{D}(\mathrm{N}(1))/\mathrm{D}(\mathrm{N}(1))$$\mathrm{a}\mathrm{i}$$\mathrm{a}\mathrm{i}$$\cdots$,,$\cdots$$\alpha_{n+1}$$\tilde{p}_{N(l)}$$\alpha_{n+1}$.

for everyis a$l$$\tilde{R}_{N(l),N(m)}$7Matrix Form of LR-Transforms and Proofof Theorem 4
Assume that the orthonormal polynomials $pN(l)$ a.1ld$=1,2,3$ ,areexpressed as linear combinations of monomials $x^{N(m)}m=1,2,3$ ,as$p_{N(l)}’l$$\cdots$$\cdots$$p_{N(l)}= \sum_{m=1}^{l}\xi_{N(l),N(m)}x^{N(m)}$47$4\ovalbox{\tt\small REJECT}_{(\mathrm{g}),N(\mathrm{m})}\mathrm{r}^{N(\cdot)}$$p\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{(\mathrm{r})}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$$771\ovalbox{\tt\small REJECT})$to be 0for $<m$ . Let —, —, be the lower
andWe putreelements $\xi N(l),N(m)$ ,triangular matrices with the $N(l)$ ,spectively. Then the orthonormality and the spectral representations for
imply the matrix relationsand$l$$\xi_{N(l),N(m)}’$$\xi_{N(l),N(m)}$$\xi_{N(l),N(m)}’$$N(m)\mathrm{t}\mathrm{h}$$A_{j}$$A_{j}’$—.

—, .

$\mathrm{C}$

(7.1)
(7.2)

$\cdot t---=1$$\mathrm{C}’\cdot t---’=1$

and
(7.3)
(7.4)

$A_{j}=---\cdot\tau_{j}^{+}\mathrm{C}\cdot t---$,$A_{j}’=---’\cdot\tau_{j}^{+}\mathrm{C}’\cdot t---$respectively.

Lemma 2Letbe the operator defined by the matr whose,,
elements are equal to 1if $(j_{1}, \cdots,j_{n})=(i_{1}$ ,and equal to 0otherwise. Then we have
$M_{\nu}$,$1\leq\nu$ $\leq n$$(i_{1}, \cdots, i_{n}; j_{1}, \cdots, j_{n})th$$1$,$i_{\nu+1}$,$\cdots$,$i_{n})$$rri,x$$\cdots$$\tau_{\nu}^{+}(\mathrm{C})=\Lambda I_{\nu}\cdot \mathrm{C}$$i_{\nu-1}$$i_{\nu}+$(7.5)

This lemma shows that (7.3) and (7.4) are equivalent to the foUowings$A_{j}=---\cdot M_{j}\cdot----1$(7.6)$A_{j}’=---’\cdot M_{j}\cdot---’-1$(7.7)

respectively. We have further$R=—\cdot---’-1$(7.8)$A_{j}’=R^{-1}\cdot A_{j}\cdot R$(7.9)(7.6)-(7.8) imply that

This proves the Theorem.
This is adiscrete analog of the argument done in [34]48
8Symmetric Polynomials Case-transforms can also be applied to symmetric orthogonal polynomials withrespect to anon-degenerate symmetric Radon measureonwithsupport which is abounded closed set.
Let , , be apartition, namely asequence of non-increasing integers$LR$$d\rho(x)$$\mathrm{R}^{n}$$\hat{D}$$\cdots$$\lambda_{1}$$\lambda_{n}$$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}\geq 0$.

Assume thatfor an incresing sequence $0<r_{1}<r_{2}<\cdots<r_{m}$ . Let
polynomials defined by the symmetrization$\lambda_{1}=\cdots=\lambda_{r_{1}}>\lambda_{\mathrm{r}+1}1=\cdots=\lambda_{f}2>\cdots>\lambda_{\mathrm{r}_{m-1}+1}=\lambda_{r_{m}}$$m_{\lambda}(x)$be the symmetric$m_{\lambda}= \frac{1}{r_{1}!(r_{2}-r_{1})!\cdots(r_{m}-r_{m-1})!}\sum_{\sigma\in \mathrm{S}_{\hslash}}\sigma(x_{1}^{\lambda_{1}}\cdots x_{n}^{\lambda_{n}})$under the permutation groupof degree .
The symmetric lexicographic ordering
can be introduced for the partitions as follows. The partitionis greater than the partthan $\mu=$
if there exists apositive integer such that
,
,
.
and
The symmetric moments axe defined as$S_{n}$$n$$\hat{O}$$\lambda=$$(\lambda_{1}, \cdots, \lambda_{n})$$(\mu_{1}, \cdots, \mu_{n})$$\mu_{1}$$\lambda_{r-1}=\mu_{f-1}$$\cdots$$\lambda_{1}=$$r$$\lambda_{f}>\mu,$$\hat{c}_{\lambda}=\frac{1}{n!}\int_{\mathrm{R}^{n}}m_{\lambda}(x)d\rho(x)$

.

(8.1)

Let
be the Hilbert space consisting of symmetric functions on
the inner product,and the norm$\hat{H}_{\rho}$$\mathrm{R}^{n}$with$||f||_{\rho}=\sqrt{(f,f)_{\rho}}$$(f, g)_{\rho}$(8.2)$(f,g)_{\rho}= \frac{1}{n!}\int_{\mathrm{R}^{n}}f(x)g(x)d\rho.(x)$for functions $f(x)$ , $g(x)$ on .
Let be the bijective mapping from the set of positive integers onto the$>\hat{N}(m.)$ for
set of all partitions such that$>m$ . Hence $N\wedge(1)=$$\hat{D}$$\hat{N}$$\hat{N}(l)$

$l$

(0,
, 0), $\hat{N}(2)=(1,0, \cdots, 0),$ $N\wedge(3)=(1,1,0, \cdots, 0)\cdots,\hat{N}(n+1)=$(1, 1,, 1), $N\wedge(n+2)=(2,0, \cdots, 0),$ $N\wedge(n+3)=(2,1,0, \cdots, 0),$ $N\wedge(n+4)=$(2, 1, 1, 0, 0), $N\wedge(2n, +1)=(2,1,1, \cdots, 1)$ , , so that we have$m_{\hat{N}(1)}(x)=1$ , $m_{\hat{N}(2)}(x)=x_{1}+\cdots+x_{n}$ ,
,
,
, etc
$\cdots$$\cdots$$\cdots$$\cdots$$m_{\dot{N}(3)}(x)=\Sigma_{1\leq i<j\leq n}x_{i}x_{j}$$x_{1}\cdots$ $x_{n}$$m_{\hat{N}(n+2)}(x)=\Sigma_{j=1}^{n}x_{j}^{2}$$m_{\hat{N}(\tau\iota+1}$

49
The generalized Hankel matrix
ments$\hat{c}_{\dot{N}(l)+\hat{N}(m)}$$\hat{\mathrm{C}}$.

are defined with the

We denote the determinants for each$\hat{N}(l)$$=\lambda$$\hat{N}(l),\hat{N}(m)\mathrm{t}\mathrm{h}$,$(8.3)$$\hat{D}_{\hat{N}(l)}=\det((\hat{c}_{\dot{N}(\mathrm{r})+\dot{N}(s)})_{\tau\cdot,s=1}^{l})$The symmetric orthogonal polynomials
tions
are given by the formulae$\tilde{\hat{p}}_{\lambda}(x)$ele-parametrized by the parti-$\hat{N}(l)=\lambda$$\tilde{\hat{p}}_{\lambda}(x)=\frac{1}{\hat{D}_{N(l-1)}}$$|\begin{array}{llll}\hat{c}_{\hat{N}(1)} \hat{c}_{\hat{N}(2)} \cdots \hat{c}_{\hat{N}(l)}\hat{c}_{\dot{N}(2)} \hat{c}_{\hat{N}(1)+\hat{N}(2)} \cdots \hat{c}_{\dot{N}(2)+\hat{N}(l)}\cdots \cdots \cdots \cdots m_{\hat{N}(\mathrm{l})}(x)\hat{c}_{\hat{N}(l-1)}\cdots \hat{c}_{\dot{N}(l-1)+\hat{N}(2)}m_{\hat{N}(2)}(x)\cdots \cdots\cdots \cdots\cdots\hat{c}_{\hat{N}(l-1)+\dot{N}(l)}m_{\hat{N}(l)}(x)\end{array}|$$=n\tau_{\hat{N}(l)}(x)+$( lower order symmetric polynomials)

(8.4)

(8.5)

The orthogonality and the norms are given by$(\tilde{\hat{p}}_{\lambda}(x),\tilde{\hat{p}}_{\mu}(x))_{\rho}$

0

$=$

$=$

(8.6)

$\lambda\neq\mu$$\lambda=\mu$$\frac{\hat{D}_{\dot{N}(l)}}{\hat{D}_{\acute{N}(l-1)}}$(8.7)

so that(8.8)$\hat{p}_{\hat{N}(l)}(x)=$are the orthonormal polynomials having the )$1$$(\hat{p}_{\lambda}(x),\hat{p}_{\mu}(x))_{\rho}$)erties$\mathrm{r}\mathrm{o}1$

I

$=$

$0$

$\lambda$$=$1

A $=\mu$

$\mu$

(8.9)

and
$p_{\hat{N}(l)}(x)=$$+\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}$order symmetric polynomials (8.10

50
Let

$e_{\mathrm{r}}$r(1$\ovalbox{\tt\small REJECT}$rz) be the elementary symmetric polynomials.
We define the bounded linear operators
Pieri$\ovalbox{\tt\small REJECT}$$e_{r}\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT} Eir_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} i_{\mathit{1}}<\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} i_{l}\ovalbox{\tt\small REJECT} ni_{\mathit{1}}i_{2}}\ovalbox{\tt\small REJECT}$

operators) on$\ovalbox{\tt\small REJECT} t^{-}t$$A_{r}$$x_{\mathrm{i}_{1}}$,:$\hat{A}_{f}$(8.11)$f(x)\in\hat{H}_{\rho}arrow e_{\mathrm{r}}(\prime x)f(x)\in\hat{H}_{\rho}$

They can be expressed in matrix form as(8.11)$e_{f}(x) \hat{p}_{\lambda}(x.)arrow\sum_{\mu}\hat{a}_{\lambda,\mu}^{(r)}\hat{p}_{\mu}(x)$Let $f(x)$ be asymmetric polynomial in , such that $f(x)$ can be expressed
as apolynomial $F$ in , ,The multiplication operator by $f(x)$ on
can be expressed as
We assume that $f(x)$ is positive in so that $F(A_{1}, \cdots, A_{n})$ is apositive
definite operator on .
Let $d\rho’(x)=f(x)d\rho(x)$ be another positive Radon measure onwith$D$
the same support as.
We denote by
the determinant. We can define
the orthogonal polynomials
andin the same way as (3.4), (3.7)
respectively.
$x$$\mathrm{e}\mathrm{i}$$e_{n}\mathrm{f}(\mathrm{x})=\mathrm{F}(\mathrm{e}\mathrm{i}, \cdots, e_{n})$$\cdots$$F(\hat{A}_{1}, \cdots,\hat{A}_{n})$$\hat{\mathcal{H}}_{\rho}$$\hat{D}$$\uparrow\{_{\rho}\wedge$$\mathrm{R}^{n}$$d\rho(x)$$\hat{D}_{\dot{N}(l)}’$$\det((\hat{c}_{\hat{N}(’\cdot)+\hat{N}(s)}’)_{\mathrm{r},s=1}^{l})$$\tilde{\hat{p}}_{\lambda}’(x)$$\acute{p}_{\lambda}’(x)$(8.11)$\hat{p}_{N(l)}’(x)=$and$(\hat{p}_{\lambda}’(x),\hat{p}_{\mu}’(x))_{d}$

$=$

$=$

0 A
1 A $=\mu$

We have the connection relations between
in the following form$\neq\mu$$\{\hat{p}_{\dot{N}(l)}(x)\}_{l\geq 1}$$\hat{p}_{\dot{N}(l)}(x)=\sum_{m=1}^{l}\hat{R}_{\hat{N}(l)/\dot{N}(m)}\hat{p}_{\dot{N}(m.)}’(x)$(8.14)

and$\{\hat{p}_{\dot{N}(l)}’(x)\}_{l\geq 1}$(8.15).

51\yen$\mathrm{j}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}(\mathrm{r}\ovalbox{\tt\small REJECT}}(x)\ovalbox{\tt\small REJECT} YI\ovalbox{\tt\small REJECT}$(8.16)$R_{j}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{(\mathrm{J})/\mathrm{p}\ovalbox{\tt\small REJECT}_{(\mathrm{m})}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{7(\mathrm{y}\mathrm{r}\mathrm{z})}}\ovalbox{\tt\small REJECT}_{\mathrm{j}^{\ovalbox{\tt\small REJECT}}}(x)$$N(\mathrm{X})$$m\ovalbox{\tt\small REJECT}$

where(8.17)$\hat{R}_{\hat{N}(l),\hat{N}(m)}=$

In particular,(8.18)$>0$$\hat{R}_{\hat{N}(l),\dot{N}(l)}=$. Let and be the corressponding lower triangularbecauseoperators which are both invertible.
$\tilde{\hat{R}}$$\tilde{\hat{R}}_{\hat{N}(l),\hat{N}(l)}=1$$\hat{R}$

Theorem 5We have the $LR$ transforms$F(\hat{A}_{1}, \cdots,\hat{A}_{n})=\hat{R}\cdot{}^{t}\hat{R}$(8.19)$\hat{A}_{r}’=\hat{R}^{-1}\cdot\hat{A}_{r}\cdot\hat{R}$(8.20)

Example 2. (Koornwinder polynomials , Heckman-Opdam
Polynomials) (see [12],[18].)

Let
$D$

:

$d\rho(x)=(1-x_{1})^{\alpha}(1-x_{2})^{\alpha}(1+x_{1})^{\beta}(1+\prime x_{2})^{\beta}(x_{1}-x_{2})^{2\gamma+1}$$-1\leq x_{2}\leq x_{2}\leq 1$

For apartition$\hat{c}_{\lambda_{1},\lambda_{2}}$$BC_{2}$-type

defined onin the 2-dimensional Euclidean space.

$\lambda_{1}\geq\lambda_{2}\geq 0$$=$$\int_{D}(x_{1}^{\lambda_{1}}x_{2^{2}}^{\lambda}+x_{2}^{\lambda_{1}}’ x_{1}^{\lambda_{2}})d\rho(x)$$=$$\int_{D}(x_{1}x_{2})^{\lambda_{1}}d\rho(x)$(8.21)$\lambda_{1}>\lambda_{2}$(8.22)$\lambda_{1}=\lambda_{2}$One can consider the symmetric orthogonal polynomials
tions of $u=x_{1}+x_{2}$ , $v=x_{1}x_{2}$ (We also denote them by

as func
or simpl$\tilde{\hat{p}}_{\hat{N}(l)}(x)$$\tilde{p}_{N(l)}^{\alpha,\beta,\gamma}(u, v)$$\mathrm{y}$52
etc according as we are interested inv),
dependence on ’\rangle t’or t.)
p is defined by the inequalities$\mathrm{b}/\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}\mathrm{K}_{(\mathrm{J})}(\mathrm{u}_{\mathrm{t}}\mathrm{v})_{\mathrm{t}}\ovalbox{\tt\small REJECT} \mathrm{L}_{(\mathrm{g})}(\mathrm{u},$$\ovalbox{\tt\small REJECT}_{N(\mathrm{x})}(\mathrm{j}_{\mathrm{t}}^{\ovalbox{\tt\small REJECT}}\mathrm{U})$$1-u+v\geq 0,1+u+v\geq 0$ ,$u^{2}-4v\geq 0$

The symmetric lexicographic orderingwith respect towith the lexicographic ordering with respect to , .
Let$\hat{O}$$\mathcal{O}$

1, , ,$u$$v$$u^{2}$, $uv$ ,$v^{2}$$u$,$u^{3}$,$u^{2}v$,$uv^{2}|$,$u^{3}$,$x_{1}$,$x_{2}$

coincides$v$$\cdots$(8.23)

be the sequence of monomials in the lexicographic order.
$\tilde{p}_{\lambda_{11}\acute{\lambda}_{2}}^{\alpha,\beta\gamma}(u, v)=u^{\lambda_{1}-\lambda_{2}}v^{\lambda_{2}}+$( lcnner order terms)

are the monic orthogonal polynomials in , , obtained by Gram-Schmidt
$u$

$v$

orthogonalization with respect to the inner product$(f(u, v)$ , $g(u, v))_{\gamma}= \int_{D}f(u, v)g(u, v)\mu^{\alpha,\beta,\gamma}(u, v)dudv$

where$\mu^{\alpha,\beta,\gamma}(u, v)$(8.24)

denotes the density$\mu^{\alpha,\beta,\gamma}(u, v)=2^{2\alpha+2\beta+2\gamma+3}(1-u+v)^{\alpha}(1+u+v)^{\beta}(u^{2}-4v)^{\gamma}$(8.25)

The first moment can be evaluated by using the. 2 dimensional Selberg
integral formula:$c_{0,0}$$=$$\hat{c}_{0,0}=2^{2\alpha+2\beta+2\gamma+2}$$\frac{\Gamma(\alpha+\gamma+\frac{3}{2})\Gamma(\beta+\gamma+\frac{3}{2})\Gamma(\alpha+1)\Gamma(\beta+1)\Gamma(2\gamma+2)}{\Gamma(\alpha+\beta+2\gamma+3)\Gamma(\alpha+\beta+\gamma+\frac{5}{2})\Gamma(\gamma+\frac{3}{2})}(8.26)$

The moments
can be expressed explicitly as follows (see [1].)We put $U=1-u+v$ and $V=1+u+v$ . Remark first that$c_{i,j}= \int_{D}u^{i}v^{j}\mu^{\alpha,\beta,\gamma}dudv$$\int_{D}U^{r}V^{s}\mu^{\alpha,\beta,\gamma}dudv=c_{0,0}\frac{(\alpha+\gamma+\frac{3}{2})_{r}(\beta+\gamma+\frac{3}{2})_{s}(\alpha+1)_{r}(\beta+1)_{s}}{(\alpha+\beta+2\gamma+3)_{r+s}(\alpha+\beta+\gamma+\frac{5}{2})_{r+s}}(8.27)$

53

where

$(\alpha)_{r}$$c_{i,j}$

$=$

$=$

denotes the product $\alpha(\alpha+1)\cdots(\alpha+r-1)$ . Hence$\int_{D}2^{-i-j}(U-V)^{i}(U+V-2)^{j}\mu^{\alpha+\beta+\gamma}dudv$$2^{-i-j}c_{0,0} \sum_{i\geq\nu_{1}\geq 0,j\geq\nu_{2}+\nu_{3}}(-1)^{j+\nu_{1}-\nu_{2}-\nu_{3}}$$( \alpha+\gamma+\frac{3}{2})_{\nu_{1}+\nu_{2}}(\beta+\gamma+\frac{3}{2})_{i-\nu_{1}+\nu_{3}}.(\alpha+1)_{\nu_{1}+\nu_{2}}(\beta+1)_{i-\nu_{1}+\nu_{3}}$$( \alpha+\beta+2\gamma+3)_{i+\nu_{2}+\nu\prime}.,(\alpha+\beta+\gamma+\frac{5}{2})_{i+\nu_{2}+\nu_{3}}$has been evaluat.ed by Heckman-Opdam in the
The norm
form of aproduct of Gamma functions. The following expression is given by
van Diejen (see ).
$(\tilde{p}_{\lambda_{1},\lambda_{2}}^{\alpha,\beta,\gamma},\tilde{p}_{\lambda_{1},\acute{\lambda}_{2}}^{\alpha,\beta\gamma})_{\gamma}$$\lfloor[8]$$(\tilde{p}_{\lambda_{1},\acute{\lambda}_{2}}^{\alpha,\beta\gamma},\tilde{p}_{\lambda_{1},\acute{\lambda}_{2}}^{\alpha,\beta\gamma})_{\gamma}=2^{2\alpha+2\beta+2\gamma+2+2(\lambda_{1}+\lambda_{2})+3}\Delta_{+}(\alpha, \beta, \gamma)\Delta_{-}(\alpha, \beta, \gamma)$(8.28)

where$\Delta_{+}(\alpha, \beta, \gamma)$$=$$\frac{\Gamma(\alpha+\beta+\gamma+\lambda_{1}+\frac{3}{2})\Gamma(\alpha+\gamma+\lambda_{1}+\frac{3}{2})\Gamma(\alpha+\beta+\lambda_{2}+1)\Gamma(\alpha+\lambda_{2}+1)}{\Gamma(\alpha+\beta+2\gamma+2\lambda_{1}+2)\Gamma(\alpha+\beta+2\lambda_{2}+1)}$.

$\frac{\Gamma(\alpha+\beta+2\gamma+\lambda_{1}+\lambda_{2}+2)\Gamma(2\gamma+\lambda_{1}-\lambda_{2}+1)}{\Gamma(\alpha+\beta+\gamma+\lambda_{1}+\lambda_{\underline{9}}+\frac{3}{2})\Gamma(\gamma+\lambda_{1}-\lambda_{2}+\underline{\frac{1}{\mathrm{Q}}})}$$\Delta_{-}(\alpha, \beta, \gamma)$$=$$\frac{\Gamma(\gamma+\lambda_{1}+\frac{3}{2})\Gamma(\beta+\gamma+\lambda_{1}+\frac{3}{2})\Gamma(\lambda_{2}+1)\Gamma(\beta+\lambda_{2}+1)}{\Gamma(\alpha+\beta+2\gamma+2\lambda_{1}+3)\Gamma(\alpha+\beta+2\lambda_{2}+2)}$$\Gamma(\alpha+\beta+\lambda_{1}+\lambda_{2}+2)\Gamma(\lambda_{1}-\lambda_{2}+1)$$\Gamma(\alpha+\beta+\gamma+\lambda_{1}+\lambda_{2}+\frac{5}{2})\Gamma(\gamma+\lambda_{1}-\lambda_{2}+\frac{\overline 3}{2})$Prom (6.7),$D_{N(l)}$is evaluated by the identities$D_{N(l)}= \prod_{m=1}^{l}(\tilde{p}_{N(m)}^{\alpha,\beta,\gamma},\tilde{p}_{N(n)}^{\alpha,\beta,\gamma^{i}},)_{\gamma}$

54

The operators$A_{1},A_{2}$and $B=A_{1}^{2}-4A_{2}$ are defined by(8.29)(8.30)(8.31)$f(u, v)arrow uf(u, v)$$f(u,v)arrow vf(u, v)$$f(u, v)arrow(u^{2}-4v)f(u, v)$

respectively.
They can be expressed in recurrence form by the use of orthogonal polynomials as follows (see $[9],[29]$ for explicit forms.):$A_{1}$$u\tilde{p}_{\lambda_{1\prime}\lambda_{2}}(x)$$=a_{\lambda_{1},\lambda_{2j}\lambda_{1}+2,\lambda_{2}}^{(1)}\tilde{p}_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}+1,\lambda_{2}}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}+2,\lambda_{2}}^{(1)}\tilde{p}_{\lambda_{1}+2,\lambda_{2}}(x)$$+a_{\lambda_{1},\lambda_{2\prime}\lambda_{1}+1,\lambda_{2}-1}^{(1)}\tilde{p}_{\lambda_{1}+1,\lambda_{2}-1}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-1,\lambda_{2}-1}^{(1)}\tilde{p}_{\lambda_{1}-1,\lambda_{2}-1}(x)$$+a_{\lambda_{1},\lambda_{2j}\lambda_{1}+1,\lambda_{2}+}^{(1)}\tilde{p}_{\lambda_{1}+1,\lambda_{2}+1}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-1,\lambda_{2}}^{(1)}\tilde{p}_{\lambda_{1}-1,\lambda_{2}}(x)$$+$$a_{\lambda_{1},\lambda_{2j}\lambda_{1}-2,\lambda_{2}}^{(1)}\tilde{p}_{\lambda_{1}-2,\lambda_{2}}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-1,\lambda_{2}+1}^{(1)}\tilde{p}_{\lambda_{1}-1,\lambda_{2}+1}(x)$:$A_{2}$(8.32)$v\tilde{p}_{\lambda_{1\prime}\lambda_{2}}(x)$$=a_{\lambda_{1},\lambda_{2j}\lambda_{1}+2,\lambda_{2}}^{(2)}\tilde{p}_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}+1,\lambda_{2}}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}+2,\lambda_{2}}^{(2)}\tilde{p}_{\lambda_{1}+2,\lambda_{2}}(x)$$+a_{\lambda_{1},\lambda_{2},\lambda_{1}+1,\lambda_{2}-1}^{(2)}\tilde{p}_{\lambda_{1}+1,\lambda_{2}-1}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-1,\lambda_{2}}^{(2)}{}_{-1}\tilde{\mathrm{P}}\lambda_{1}-1,\lambda_{2}-1(x)$$+a_{\lambda_{1},\lambda_{2j}\lambda_{1}+1,\lambda_{2}+}^{(2)}\tilde{p}_{\lambda_{1}+1,\lambda_{2}+1}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-1,\lambda_{2}}^{(2)}\tilde{p}_{\lambda_{1}-1,\lambda_{2}}(x)$$+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-2,\lambda_{2}}^{(2)}\tilde{p}_{\lambda_{1}-2,\lambda_{2}}(x)+a_{\lambda_{1},\lambda_{2j}\lambda_{1}-1,\lambda_{2}+1}^{(2)}\tilde{p}_{\lambda_{1}-1,\lambda_{2}+1}(x)$$B$:(8.33)$(u^{2}-4v)\tilde{p}_{\lambda_{1\prime}\lambda_{2}}(x)$$=b_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}+2,\lambda_{2}}\tilde{p}_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}+2,\lambda_{2}}(x)+b_{\lambda_{1},\lambda_{2j}\lambda_{1}+1,\lambda_{2}}\tilde{p}_{\lambda_{1}+1,\lambda_{2}}(x)$$+b_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}+1,\lambda_{2}-1}\tilde{p}_{\lambda_{1}+1,\lambda_{2}-1}(x)+b_{\lambda_{1\prime}\lambda_{2}.\lambda_{1}-1,\lambda_{2}-1}.\tilde{p}_{\lambda_{1}-1,\lambda_{2}-1}(x)$$+b_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}+1,\lambda_{\sim}+1},\tilde{p}_{\lambda_{1}+1,\lambda_{2}+1}(x)+b_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}-1,\lambda_{2}}\tilde{p}_{\lambda_{1}-1,\lambda_{2}}(x)$$+b_{\lambda_{1\prime}\lambda_{2j}\lambda_{1}-2,\bigwedge_{2}\tilde{p}_{\lambda_{1}-2,\lambda_{2}}}(X)+b_{\lambda_{1\prime}\lambda_{21}\lambda_{1}-1,\lambda_{2}+1\tilde{p}_{\lambda_{1}-1,\lambda_{2}+1}}.(\prime x)$,
where the matrices
bounded self-adjoint operators Ai,$(a_{\lambda_{1},\lambda_{2j}\mu_{1},\mu 2}^{(1)})$$(a_{\lambda_{1},\lambda_{2j}\mu_{1},\mu_{2}}^{(2)})$$A_{2}$

,

$B$

on

, and

define therespectively.

$(b_{\lambda_{1},\lambda_{2j}\mu_{1},\mu_{2}})$$L^{2}(\mathrm{Z}_{\geq 0}^{2})$(8.34)

55
We have the connection formulae between the \yen and\yen follows.
$=*I\rangle$$,1_{\mathrm{t}}’ 2$$\mathrm{A}_{1_{\mathrm{t}^{\ovalbox{\tt\small REJECT}}}}\mathrm{X}_{2}$$\tilde{p}_{\lambda_{1\prime}\lambda_{2}}^{\gamma}(x)=\tilde{p}_{\lambda_{1\prime}\lambda_{2}}^{\gamma+1}(x)+\tilde{R}_{\lambda_{1\prime}\lambda_{2}/\lambda_{1}-1,\lambda_{2}-1}\tilde{p}_{\lambda_{1}-1,\lambda_{2}-1}^{\gamma+1}(x)$$+$$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}+1}\tilde{p}_{\lambda_{1}-1,\lambda_{2}+1}^{\gamma+1}(x)$$+$$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}}\tilde{p}_{\lambda_{1}-1,\lambda_{2}}^{\gamma+1}(x)+\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-2,\lambda_{2}}\tilde{p}_{\lambda_{1}-2,\lambda_{2}}^{\gamma+1}(x)$(8.35). We denote by theWe putto be 0forlower triangular matrix
thus obtained.
The matrix is then defined by (6.19).
, $B’$ be the corresponding operators for
Let the operators A[,
in place of . Then the following formulae of-transforms similar to (6.19)
and (6.20) hold.
$(\lambda_{1}, \lambda_{2})<(\mu_{1}, \mu_{2})$$\tilde{R}_{\lambda_{1},\lambda_{2}/\mu_{1},\mu_{2}}$$\tilde{R}$$\{\tilde{R}_{\lambda_{1},\lambda_{2}/\mu_{1},\mu_{2}}\}_{\lambda_{1},\lambda_{2}/\mu_{1},\mu_{2}}$$R$$\wedge/+1$$A_{2}’$$\mathrm{L}\mathrm{R}$$\gamma$$B=R\cdot {}^{t}R$$A_{1}’=R^{-1}\cdot A_{1}\cdot R$$A_{2}’=R^{-1}\cdot A_{2}\cdot R$(8.36)(8.37)(8.38)

These are equivalent to the idetities (8.18),(8.19) and (8.20) respectively.
The elements of can be evaluated in aremarkable way.
$\tilde{R}$Proposition 8$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}+1}$$=$$- \frac{(\lambda_{1}-\lambda_{2}-1)(\lambda_{1}-\lambda_{2})}{(\gamma+\lambda_{1}-\lambda_{2}-\frac{1}{2})(\gamma+\lambda_{1}-\lambda_{2}+\frac{1}{2})}$$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}}$$4(\lambda_{1}-\lambda_{2})(\alpha-\beta)(\alpha+\beta)(\alpha+\beta+\lambda_{1}+\lambda_{2}+1)$$(\alpha+\beta+2\lambda_{2})(\alpha+\beta+2\lambda_{2}+2)(\alpha+\beta+2\gamma+2\lambda_{1}+1)(\alpha+\beta+2\gamma+2\lambda_{1}+1)$$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}-1}$$=$$- \frac{4\lambda_{2}(\alpha+\lambda_{2})(\beta+\lambda_{2})(\alpha+\beta+\lambda_{2})}{(\alpha+\beta+\gamma+\lambda_{1}+\lambda_{2}+\underline{\frac{1}{9}})(\alpha+\beta+\gamma+\lambda_{1}+\lambda_{2}+\frac{3}{2})}$$(\alpha+\beta+\lambda_{1}+\lambda_{2})(\alpha+\beta+\lambda_{1}+\lambda_{2}+1)$$(\alpha+\beta+2\lambda_{2}-1)(\alpha+\beta+2\lambda_{2})^{2}(\alpha+\beta+2\lambda_{2}+1)$

56$R_{\lambda_{1\prime}\lambda_{2}/\lambda_{1}-2,\lambda_{2}}$$=$$\frac{4(\lambda_{1}-\lambda_{2}-1)(\lambda_{1}-\lambda_{2})(\alpha+\sqrt+\lambda_{1}+\lambda_{2})(\alpha+\sqrt+\lambda_{1}+\lambda_{2}+1)}{(\gamma+\lambda_{1}-\lambda_{2}-\frac{1}{2})(\gamma+\lambda_{1}-\lambda_{2}+\frac{1}{2})(\alpha+\beta+\gamma+\lambda_{1}+\lambda_{2}+\frac{1}{2})(\alpha+\sqrt+\gamma+)}$$\frac{(\alpha+\gamma+\lambda_{1}+\frac{1}{2})(\sqrt+\gamma+\lambda_{1}+\frac{1}{2})(\gamma+\lambda_{1}+\frac{1}{2})(\alpha+\sqrt+\gamma+\lambda_{1}+\frac{1}{2})}{(\alpha+\sqrt+2\gamma+2\lambda_{1})(\alpha+\sqrt+2\gamma+2\lambda_{1}+1)^{2}(\alpha+\sqrt+2\gamma+2\lambda_{1}+2)}$

The operator $B_{+}=1-A_{1}+A_{2}$ corresponding to the shiftdefined by$\alphaarrow\alpha+1$is$B_{+}:$ $\varphi(u, v)arrow(1-u+v)\varphi(\tau\iota, v)$

Its connection formula foris given by$\tilde{p}_{\lambda_{1},\acute{\lambda}_{2}}^{\alpha,\beta\gamma}(x)=\tilde{p}_{\lambda_{1},\lambda_{2}}^{\alpha}(x)$and$\tilde{p}_{\lambda_{1},\lambda_{2}}^{\alpha+1,\beta,\gamma}(x)=\tilde{p}_{\lambda_{1},\lambda_{2}}^{\alpha+1}(x)$$\tilde{p}_{\lambda_{1\prime}\lambda_{2}}^{\alpha}(x)=\tilde{p}_{\lambda_{1},\lambda_{2}}^{\alpha+1}(x)+\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1},\lambda_{2}-1}\tilde{p}_{\lambda_{1},\lambda_{2}-1}^{\alpha+1}(x)$$+$$\tilde{R}_{\lambda_{1\prime}\lambda_{2}/\lambda_{1}-1,\lambda_{2}}\tilde{p}_{\lambda_{1}-1,\lambda_{2}}^{\alpha+1}(x)+\tilde{R}_{\lambda_{1\prime}\lambda_{2}/\lambda_{1}-1,\lambda_{2}-1}\tilde{p}_{\lambda_{1}-1,\lambda_{2}-1}^{\alpha+1}(x)$where$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1},\lambda_{2}-1}=-2\frac{\lambda_{2}(\sqrt+\lambda_{2})}{(\alpha+\sqrt+2\lambda_{2})(\alpha+\sqrt+2\lambda_{2}+1)}$$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}+1}$$=$$-2 \frac{(\lambda_{1}-\lambda_{2})(2\gamma+\lambda_{1}-\lambda_{2})}{(\gamma+\lambda_{1}-\lambda_{2}+\frac{1}{2})(\gamma+\lambda_{1}-\lambda_{2}-\frac{1}{2})}$$\frac{(\gamma+\lambda_{1}+\frac{1}{2})(\sqrt+\gamma+\lambda_{1}+\frac{1}{2})}{(\alpha+\sqrt+2\gamma+2\lambda_{1}+1)(\alpha+\sqrt+2\gamma+2\lambda_{1}+2)}$$\tilde{R}_{\lambda_{1},\lambda_{2}/\lambda_{1}-1,\lambda_{2}-1}$$=$$4 \frac{\lambda_{2}(\sqrt+\lambda_{2})(\gamma+\frac{1}{2})}{(\alpha+\sqrt+2\lambda_{2})(\alpha+\sqrt+2\lambda_{2}+1)(\alpha+\sqrt+2\gamma+2\lambda_{1}+1)}$.
$\frac{(\sqrt+\gamma+\lambda_{1}+\frac{1}{2})(\alpha+\sqrt+2\gamma+\lambda_{1}+\lambda_{2}+1)(\alpha+\sqrt+\lambda_{1}+\lambda_{2}+1)}{(\alpha+\sqrt+2\gamma+2\lambda_{1}+2)(\alpha+\sqrt+\gamma+\lambda_{1}+\lambda_{2}+\frac{1}{2})(\alpha+\sqrt+\gamma+\lambda_{1}+\lambda_{2}+\frac{3}{2})}$

One can obtain asimilar formula for the operator $B_{-}=1+A_{1}+A_{2}$.
, seeing thatinduced by the shift$\sqrtarrow\sqrt+1$$\tilde{p}_{\lambda_{1},\acute{\lambda}_{2}}^{\alpha,\beta\gamma}(x)=(-1)^{\lambda_{1}+\lambda_{2}}\tilde{p}_{\lambda_{1},\acute{\lambda}_{2}}^{\theta,\alpha\gamma}(-x)$

57
We have the corresponding-transforms$\mathrm{L}/\mathrm{Z}$

1% $A_{1}+A_{2}=R\cdot {}^{t}R$ ,$A_{1}’=R^{-1}\cdot A_{1}\cdot R$$A_{2}’=R^{-1}\cdot A_{2}\cdot R$

respectively.
Details of these formulae will be discussed later.

Remark 3It seems interesting to generalize our obser vations to more general case. For example, Koornwinder polynbomials have been generalized to$BC$-type orthogonal polynomials by Heckman-Opdam.
Prof. K. Kadell has
discussed the connection relations (8.15)-(8.16) in the case of Selberg-Jack
polynomials. One may ask if simple product formulae like in Proposition 8
will be given in these cases.

References[1] G.Andrews, R.Askey and R.Roy.

2000.

Special Functions, Cambridge UP.

[2] K.Aomoto. Density matrices for Eigenfunction Expansions and LRTransforms, preprint, 2000.

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Acad., 61, Ser.A, Nol(1985) 11-14; 157-160.
[4] K.Aomoto, Self-adjointness and limit pointness for adjacency operators
on a tree, J. d’Anal. Math., 53(1989), 219-232.

[5] R.Askey and J.A.Wilson Some basic hypergeometric orthogonal polynOmials that generalize Jacobi polynomials, Mem. Amer. Math. Soc.,1985..
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[15] K.Kadell. The Selberg-Jack Symmetric Functions,130(1997), 33-102.

Ad in Math.,[16] K.Kadell. The Dyson Polynomials, mimeographed note, 2000.

[17] J.Kaneko. Selberg Integrals and Hypergeometric Functions Associatedwith Jack Polynomials, SIAM J. Math. Anal., 24(1993), 1086-1110.
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59[22] A.Mukaihira and Y.Nakamura, Integrable discretization of the modified
KdV equations and $applications_{f}$ Inverse Problems, 16(2000), 413-424.
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[27] H.Rutishauser, Une methode pour la deter mination des valeurs propresd’une matrice, C.R.Acad.Sci.Paris 240(1955), 34-36[28] Jiro Sekiguchi. Zonal Spherical Functions on Some Symmetric Spaces
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