Hypergeometric function of a matrix argument

Definition

Let p\ge 0 and q\ge 0 be integers, and let X be an m\times m complex symmetric matrix. Then the hypergeometric function of a matrix argument X and parameter \alpha0 is defined as

: pF_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)^{(\alpha )}\kappa\cdots(a_p)\kappa^{(\alpha )}} {(b_1)\kappa^{(\alpha )}\cdots(b_q)\kappa^{(\alpha )}} \cdot C_\kappa^{(\alpha )}(X),

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

Two matrix arguments

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

: pF_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)^{(\alpha )}\kappa\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 I is the identity matrix of size m.

Not a 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 ''α''

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

The thing to remember is that

: \alpha=\frac{2}{\beta}.

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

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