Hypergeometric function of a matrix argument: Difference between revisions

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In [[mathematics]], the '''hypergeometric function of a matrix argument''' is a generalization of the classical [[hypergeometric series]]. It is the closed form expression of certain multivariate integrals, especially ones appearing 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==

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and parameter <math>\alpha>0</math> is defined as

: <math>

<center>

<math>

_pF_q^{(\alpha )}(a_1,\ldots,a_p;

b_1,\ldots,b_q;X) =

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C_\kappa^{(\alpha )}(X),

</math>

</center>

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

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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 argument is defined as:

: <math>

<center>

<math>

_pF_q^{(\alpha )}(a_1,\ldots,a_p;

b_1,\ldots,b_q;X,Y) =

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}{C_\kappa^{(\alpha )}(I)},

</math>

</center>

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

==Not youra 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'''!

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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==

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* Robb Muirhead, ''Aspects of Multivariate Statistical Theory'', John Wiley & Sons, Inc., New York, 1984.

==External links==