Hypergeometric function of a matrix argument
Encyclopedia
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.
and 
be integers, and let
be an 
complex symmetric matrix.
Then the hypergeometric function of a matrix argument
and parameter
is defined as
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series
Hypergeometric series
In mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by...
. 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.
Definition
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 partitionPartition (number theory)In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...
of
, 
is the Generalized Pochhammer symbol, and
is the ``C" normalization of the Jack functionJack functionIn mathematics, the Jack function, introduced by Henry Jack, is a homogenous, symmetric polynomial which generalizes the Schur and zonal polynomials,and is in turn generalized by the Heckman–Opdam polynomials and Macdonald polynomials.-Definition:...
.
Two matrix arguments
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 argument
Unlike other functions of matrix argument, such as the matrix exponentialMatrix exponentialIn mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....
, which are matrix-valued, the hypergeometric function of (one or two) matrix arguments is scalar-valued.
The parameter
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 
.
External links
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