Hypergeometric function of a matrix argument

Definition

Let p ≥ 0 and q ≥ 0 be integers, and let X be an m × m complex symmetric matrix. Then the hypergeometric function of a matrix argument X and parameter α > 0 is defined as

where κ ⊢ k means κ is a partition of k , ( a i ) κ ( α ) is the Generalized Pochhammer symbol, and C κ ( α ) ( X ) is the "C" normalization of the Jack function.

Two matrix arguments

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

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 {\displaystyle \alpha }

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

The thing to remember is that

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

Typically, in settings involving real random matrices, α = 2 and thus β = 1 . In settings involving complex random matrices, one has α = 1 and β = 2 .