Английская Википедия: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 <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 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.
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