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 the closed form expression of certain multivariate integrals, especially ones appearing 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 ${\displaystyle p\geq 0}$  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 (number theory) 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.

Two matrix arguments

If ${\displaystyle X}$  and ${\displaystyle Y}$  are two ${\displaystyle m\times m}$  complex symmetric matrices, then the hypergeometric function of two matrix argument 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 ${\displaystyle m}$ .

Not your typical function of a matrix argument

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!

The parameter ${\displaystyle \alpha }$

In many publications the parameter ${\displaystyle \alpha }$  is omitted yet in different publications different values of ${\displaystyle \alpha }$  are being implicitly assumed. For example, in the theory of real random matrices (see, e.g., Muirhead, 1984) ${\displaystyle \alpha =2}$  whereas in other settings (e.g., in the complex case--see Gross and Richards, 1989) ${\displaystyle \alpha =1}$ . To make matters worse, in combinatorics researchers often use the parameter ${\displaystyle \alpha }$  whereas in random matrix theory reserchers tend to prefer a parameter called ${\displaystyle \beta <math>.Inotherdisciplines,itis<math>\alpha /2}$  that has certain meaning. In either case the connection is simple:

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

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

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

References

• 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.

• Kaneko, J., "Selberg Integrals and hypergeometric functions associated with Jack polynomials", SIAM Journal on Mathematical Analysis, 24, no. 4, 1086-1110, 1993.

• Koev, Plamen and Edelman, Alan, "The efficient evaluation of the hypergeometric function of a matrix argument", Mathematics of Computation, 75, no. 254, 833-846, 2006.

• Muirhead, Robb, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.

External links

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