Hypergeometric function of a matrix argument

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.

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

${\displaystyle _{p}F_{q}^{(\alpha )}(a_{1},\ldots ,a_{p};b_{1},\ldots ,b_{q};X)=\sum _{k=0}^{\infty }\sum _{\kappa \vdash k}{\frac {1}{k!}}\cdot {\frac {(a_{1})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot C_{\kappa }^{(\alpha )}(X),}${\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 ${\displaystyle \kappa \vdash k}${\displaystyle \kappa \vdash k} means ${\displaystyle \kappa }${\displaystyle \kappa } is a partition of ${\displaystyle k}${\displaystyle k}, ${\displaystyle (a_{i})_{\kappa }^{(\alpha )}}${\displaystyle (a_{i})_{\kappa }^{(\alpha )}} is the generalized Pochhammer symbol, and ${\displaystyle C_{\kappa }^{(\alpha )}(X)}${\displaystyle C_{\kappa }^{(\alpha )}(X)} is the "C" normalization of the Jack function.

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

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

In many publications the parameter ${\displaystyle \alpha }${\displaystyle \alpha } is omitted. Also, in different publications different values of ${\displaystyle \alpha }${\displaystyle \alpha } are being implicitly assumed. For example, in the theory of real random matrices (see, e.g., Muirhead, 1984), ${\displaystyle \alpha =2}${\displaystyle \alpha =2} whereas in other settings (e.g., in the complex case—see Gross and Richards, 1989), ${\displaystyle \alpha =1}${\displaystyle \alpha =1}. To make matters worse, in random matrix theory researchers tend to prefer a parameter called ${\displaystyle \beta }${\displaystyle \beta } instead of ${\displaystyle \alpha }${\displaystyle \alpha } which is used in combinatorics.

The thing to remember is that

${\displaystyle \alpha ={\frac {2}{\beta }}.}${\displaystyle \alpha ={\frac {2}{\beta }}.}

Care should be exercised as to whether a particular text is using a parameter ${\displaystyle \alpha }${\displaystyle \alpha } or ${\displaystyle \beta }${\displaystyle \beta } and which the particular value of that parameter is.

Typically, in settings involving real random matrices, ${\displaystyle \alpha =2}${\displaystyle \alpha =2} and thus ${\displaystyle \beta =1}${\displaystyle \beta =1}. In settings involving complex random matrices, one has ${\displaystyle \alpha =1}${\displaystyle \alpha =1} and ${\displaystyle \beta =2}${\displaystyle \beta =2}.

• K. I. Gross and D. St. P. Richards, "Total positivity, spherical series, and hypergeometric functions of matrix argument", J. Approx. Theory, 59, no. 2, 224–246, 1989.
• J. Kaneko, "Selberg Integrals and hypergeometric functions associated with Jack polynomials", SIAM Journal on Mathematical Analysis, 24, no. 4, 1086-1110, 1993.
• Plamen Koev and Alan Edelman, "The efficient evaluation of the hypergeometric function of a matrix argument", Mathematics of Computation, 75, no. 254, 833-846, 2006.
• Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.

• Software for computing the hypergeometric function of a matrix argument by Plamen Koev.

• Hypergeometric functions