Package: pyhypergeomatrix
pyhypergeomatrixHypergeometric functions of a matrix argument.Evaluation of the hypergeometric function of a matrix argument (Koev & Edelman's algorithm)Let $(a_1, \ldots, a_p)$ and $(b_1, \ldots, b_q)$ be two vectors of real or
complex numbers, possibly empty, $\alpha > 0$ and $X$ a real symmetric or a
complex Hermitian matrix.
The correspondinghypergeometric function of a matrix argumentis defined by$${}_pF_q^{(\alpha)} \left(\begin{matrix} a_1, \ldots, a_p \\ b_1, \ldots, b_q\end{matrix}; X\right) = \sum_{k=0}^{\infty}\sum_{\kappa \vdash k} \frac{{(a_1)}_{\kappa}^{(\alpha)} \cdots {(a_p)}_{\kappa}^{(\alpha)}} {{(b_1)}_{\kappa}^{(\alpha)} \cdots {(b_q)}_{\kappa}^{(\alpha)}} \frac{C_{\kappa}^{(\alpha)}(X)}{k!}.$$The inner sum is over the integer partitions $\kappa$ of $k$ (which we also
denote by $|\kappa| = k$). The symbol ${(\cdot)}_{\kappa}^{(\alpha)}$ is thegeneralized Pochhammer symbol, defined by$${(c)}^{(\alpha)}_{\kappa} = \prod_{i=1}^{\ell}\prod_{j=1}^{\kappa_i} \left(c - \frac{i-1}{\alpha} + j-1\right)$$when $\kappa = (\kappa_1, \ldots, \kappa_\ell)$.
Finally, $C_{\kappa}^{(\alpha)}$ is aJack function.
Given an integer partition $\kappa$ and $\alpha > 0$, and a
real symmetric or complex Hermitian matrix $X$ of order $n$,
the Jack function$$C_{\kappa}^{(\alpha)}(X) = C_{\kappa}^{(\alpha)}(x_1, \ldots, x_n)$$is a symmetric homogeneous polynomial of degree $|\kappa|$ in the
eigen values $x_1$, $\ldots$, $x_n$ of $X$.The series defining the hypergeometric function does not always converge.
See the references for a discussion about the convergence.The inner sum in the definition of the hypergeometric function is over
all partitions $\kappa \vdash k$ but actually
$C_{\kappa}^{(\alpha)}(X) = 0$ when $\ell(\kappa)$, the number of non-zero
entries of $\kappa$, is strictly greater than $n$.For $\alpha=1$, $C_{\kappa}^{(\alpha)}$ is aSchur polynomialand it is
azonal polynomialfor $\alpha = 2$.
In random matrix theory, the hypergeometric function appears for $\alpha=2$
and $\alpha$ is omitted from the notation, implicitely assumed to be $2$.Koev and Edelman (2006) provided an efficient algorithm for the evaluation
of the truncated series$$\sideset{_p^m}{_q^{(\alpha)}}F \left(\begin{matrix} a_1, \ldots, a_p \\ b_1, \ldots, b_q\end{matrix}; X\right) = \sum_{k=0}^{m}\sum_{\kappa \vdash k} \frac{{(a_1)}_{\kappa}^{(\alpha)} \cdots {(a_p)}_{\kappa}^{(\alpha)}} {{(b_1)}_{\kappa}^{(\alpha)} \cdots {(b_q)}_{\kappa}^{(\alpha)}}
\frac{C_{\kappa}^{(\alpha)}(X)}{k!}.$$Hereafter, $m$ is called thetruncation weight of the summation(because $|\kappa|$ is called the weight of $\kappa$), the vector
$(a_1, \ldots, a_p)$ is called the vector ofupper parameterswhile
the vector $(b_1, \ldots, b_q)$ is called the vector oflower parameters.
The user has to supply the vector $(x_1, \ldots, x_n)$ of the eigenvalues
of $X$.For example, to compute$$\sideset{_2^{15}}{_3^{(2)}}F \left(\begin{matrix} 3, 4 \\ 5, 6, 7\end{matrix}; 0.1, 0.4\right)$$you have to enter>>>frompyhypergeomatrix.hypergeomatimporthypergeomPQ>>>hypergeomPQ(15,[3,4],[5,6,7],[0.1,0.4],2)We said that the hypergeometric function is defined for a real symmetric
matrix or a complex Hermitian matrix $X$. Thus the eigenvalues of $X$
are real. However we do not impose this restriction inpyhypergeomatrix.
The user can enter any list of real or complex numbers for the eigenvalues.Univariate caseFor $n = 1$, the hypergeometric function of a matrix argument is known as thegeneralized hypergeometric function.
It does not depend on $\alpha$. The case of $\sideset{_{2\thinspace}^{}}{_1^{}}F$ is the most known,
this is the Gauss hypergeometric function. Let's check a value. It is known that$$\sideset{_{2\thinspace}^{}}{_1^{}}F \left(\begin{matrix} 1/4, 1/2 \\ 3/4\end{matrix}; 80/81\right) = 1.8.$$Since $80/81$ is close to $1$, the convergence is slow. We compute the truncated series below
for $m = 300$.>>>frompyhypergeomatrix.hypergeomatimporthypergeomPQ>>>hypergeomPQ(300,[1/4,1/2],[3/4],[80/81])1.79900265281923ReferencesPlamen Koev and Alan Edelman.The efficient evaluation of the hypergeometric function of a matrix argument.
Mathematics of computation, vol. 75, n. 254, 833-846, 2006.Robb Muirhead.Aspects of multivariate statistical theory.
Wiley series in probability and mathematical statistics.
Probability and mathematical statistics.
John Wiley & Sons, New York, 1982.A. K. Gupta and D. K. Nagar.Matrix variate distributions.
Chapman and Hall, 1999.