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In [[mathematics]], the '''hypergeometric function of a matrix argument''' is a generalization of the classical [[hypergeometric series]]. It is
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
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and parameter <math>\alpha>0</math> is defined as
: <math>▼
▲<math>
_pF_q^{(\alpha )}(a_1,\ldots,a_p;
b_1,\ldots,b_q;X) =
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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 [[
<math>C_\kappa^{(\alpha )}(X)</math> is the
==Two matrix arguments==
==References==▼
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>
* 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.▼
_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>.
* Koev, Plamen and Edelman, Alan, "The efficient evaluation of the hypergeometric function of a matrix argument", ''Mathematics of Computation'', '''75''', no. 254, 833-846, 2006.▼
==Not a typical function of a matrix argument==
* Muirhead, Robb, ''Aspects of Multivariate Statistical Theory'', John Wiley & Sons, Inc., New York, 1984.▼
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.
▲*
▲*
==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)}}
▲* [http://www-math.mit.edu/~plamen/software/mhgref.html Software for computing the hypergeometric function of a matrix argument]
[[Category:Hypergeometric functions]]
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