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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 <math>p\ge 0</math> and <math>q\ge 0</math> be integers, and let
<math>X </math> be an <math>m\times m</math> complex symmetric matrix.
Then the hypergeometric function of a matrix argument <math>X</math>
and parameter <math>\alpha>0</math> is defined as

: <math>
_pF_q^{(\alpha )}(a_1,\ldots,a_p;
b_1,\ldots,b_q;X) =
\sum_{k=0}^\infty\sum_{\kappa\vdash k}
\frac{1}{k!}\cdot
\frac{(a_1)^{(\alpha )}_\kappa\cdots(a_p)_\kappa^{(\alpha )}}
{(b_1)_\kappa^{(\alpha )}\cdots(b_q)_\kappa^{(\alpha )}} \cdot
C_\kappa^{(\alpha )}(X),
</math>

where <math>\kappa\vdash k</math> means <math>\kappa</math> is a [[partition (number theory)|partition]] of <math>k</math>, <math>(a_i)^{(\alpha )}_{\kappa}</math> is the [[generalized Pochhammer symbol]], and
<math>C_\kappa^{(\alpha )}(X)</math> is the "C" normalization of the [[Jack function]].

==Two matrix arguments==
If <math>X</math> and <math>Y</math> are two <math>m\times m</math> complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:

: <math>
_pF_q^{(\alpha )}(a_1,\ldots,a_p;
b_1,\ldots,b_q;X,Y) =
\sum_{k=0}^\infty\sum_{\kappa\vdash k}
\frac{1}{k!}\cdot
\frac{(a_1)^{(\alpha )}_\kappa\cdots(a_p)_\kappa^{(\alpha )}}
{(b_1)_\kappa^{(\alpha )}\cdots(b_q)_\kappa^{(\alpha )}} \cdot
\frac{C_\kappa^{(\alpha )}(X)
C_\kappa^{(\alpha )}(Y)
}{C_\kappa^{(\alpha )}(I)},
</math>

where <math>I</math> is the identity matrix of size <math>m</math>.

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

The thing to remember is that

: <math>\alpha=\frac{2}{\beta}.</math>

Care should be exercised as to whether a particular text is using a parameter <math>\alpha</math> or <math>\beta</math> and which the particular value of that parameter is.

Typically, in settings involving real random matrices, <math>\alpha=2</math> and thus <math>\beta=1</math>. In settings involving complex random matrices, one has <math>\alpha=1</math> and <math>\beta=2</math>.

==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==
* [http://www-math.mit.edu/~plamen/software/mhgref.html Software for computing the hypergeometric function of a matrix argument] by Plamen Koev.

{{series (mathematics)}}
[[Category:Hypergeometric functions]]

Hypergeometric function of a matrix argument - Revision history

imported>ReGuess: /* The parameter \alpha */ WP:SECTIONHEAD: For technical reasons, section headings should not contain markup. (I don't know if this is the preferred way of fixing this. If not, feel free to improve this further!)

imported>ReGuess: /* The parameter \alpha */ WP:SECTIONHEAD: For technical reasons, section headings should not contain $markup. (I don't know if this is the preferred way of fixing this. If not, feel free to improve this further!)$