Hypergeometric function of a matrix argument

Let ${\displaystyle p\geq 0}$ and ${\displaystyle q\geq 0}$ be integers, and let ${\displaystyle X}$ be an ${\displaystyle m\times m}$ complex symmetric matrix. Then the hypergeometric function of a matrix argument ${\displaystyle X}$ and parameter ${\displaystyle \alpha >0}$ is defined as

${\displaystyle _{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot C_{\kappa }^{(\alpha )}(X),}$

where ${\displaystyle \kappa \vdash k}$ means ${\displaystyle \kappa }$ is a partition of ${\displaystyle k}$, ${\displaystyle (a_{i})_{\kappa }^{(\alpha )}}$ is the generalized Pochhammer symbol, and ${\displaystyle C_{\kappa }^{(\alpha )}(X)}$ is the "C" normalization of the Jack function.

If ${\displaystyle X}$ and ${\displaystyle Y}$ are two ${\displaystyle m\times m}$ complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:

${\displaystyle _{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X,Y)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot {\frac {C_{\kappa }^{(\alpha )}(X)C_{\kappa }^{(\alpha )}(Y)}{C_{\kappa }^{(\alpha )}(I)}},}$

where ${\displaystyle I}$ is the identity matrix of size ${\displaystyle m}$.

Unlike other functions of matrix argument, such as the matrix exponential, which are matrix-valued, the hypergeometric function of (one or two) matrix arguments is scalar-valued.