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.
complex symmetric matrix.
Then the hypergeometric function of a matrix argument
is the generalized Pochhammer symbol, and
is the "C" normalization of the Jack function.
complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as: where
is the identity matrix of size
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
Also, in different publications different values of
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
The thing to remember is that Care should be exercised as to whether a particular text is using a parameter
and which the particular value of that parameter is.
Typically, in settings involving real random matrices,
In settings involving complex random matrices, one has