Hypergeometric function of a matrix argument

Summary

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.

DefinitionEdit

Let   and   be integers, and let   be an   complex symmetric matrix. Then the hypergeometric function of a matrix argument   and parameter   is defined as

 

where   means   is a partition of  ,   is the generalized Pochhammer symbol, and   is the "C" normalization of the Jack function.

Two matrix argumentsEdit

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

 

where   is the identity matrix of size  .

Not a typical function of a matrix argumentEdit

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 αEdit

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

Typically, in settings involving real random matrices,   and thus  . In settings involving complex random matrices, one has   and  .

ReferencesEdit

  • 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.

External linksEdit

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