Hypergeometric Function Of A Matrix Argument

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.

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 $\alpha>0$ is defined as

: $_pF_q^(a_1,\ldots,a_p; b_1,\ldots,b_q;X) = \sum_^\infty\sum_ \frac\cdot \frac \cdot C_\kappa^(X),$

where $\kappa\vdash k$ means $\kappa$ is a partition of $k$, $(a_i)^_$ is the generalized Pochhammer symbol, and
$C_\kappa^(X)$ 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:

: $_pF_q^(a_1,\ldots,a_p; b_1,\ldots,b_q;X,Y) = \sum_^\infty\sum_ \frac\cdot \frac \cdot \frac,$

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

: $\alpha=\frac.$

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.

External links

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

{{series (mathematics)
Hypergeometric functions