In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to evaluate certain multivariate integrals. Hypergeometric functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are often expressed in terms of the hypergeometric function of a matrix argument.
Definition Let p ≥ 0 {\displaystyle p\geq 0} and q ≥ 0 {\displaystyle q\geq 0} be integers, and let
X {\displaystyle X} be an m × m {\displaystyle m\times m} complex symmetric matrix. Then the hypergeometric function of a matrix argument X {\displaystyle X}
and parameter α > 0 {\displaystyle \alpha >0} is defined as
p F q ( α ) ( a 1 , … , a p ; b 1 , … , b q ; X ) = ∑ k = 0 ∞ ∑ κ ⊢ k 1 k ! ⋅ ( a 1 ) κ ( α ) ⋯ ( a p ) κ ( α ) ( b 1 ) κ ( α ) ⋯ ( b q ) κ ( α ) ⋅ C κ ( α ) ( X ) , {\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 κ ⊢ k {\displaystyle \kappa \vdash k} means κ {\displaystyle \kappa } is a partition of k {\displaystyle k} , ( a i ) κ ( α ) {\displaystyle (a_{i})_{\kappa }^{(\alpha )}} is the generalized Pochhammer symbol, and
C κ ( α ) ( X ) {\displaystyle C_{\kappa }^{(\alpha )}(X)} is the "C" normalization of the Jack function.
Two matrix arguments If X {\displaystyle X} and Y {\displaystyle Y} are two m × m {\displaystyle m\times m} complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:
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