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

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Let  p ≥ 0 {\displaystyle p\geq 0} and  q ≥ 0 {\displaystyle q\geq 0} be integers, and let  X {\displaystyle X} be an  m × m {\displaystyle m\times m} complex symmetric matrix. Then the hypergeometric function of a matrix argument  X {\displaystyle X} and parameter  α > 0 {\displaystyle \alpha >0} is defined as

p F q ( α ) ( a 1 , … , a p ; b 1 , … , b q ; X ) = ∑ k = 0 ∞ ∑ κ ⊢ k 1 k ! ⋅ ( a 1 ) κ ( α ) ⋯ ( a p ) κ ( α ) ( b 1 ) κ ( α ) ⋯ ( b q ) κ ( α ) ⋅ C κ ( α ) ( X ) , {\displaystyle _{p}F_{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})_{\kappa }^{(\alpha )}\cdots (a_{p})_{\kappa }^{(\alpha )}}{(b_{1})_{\kappa }^{(\alpha )}\cdots (b_{q})_{\kappa }^{(\alpha )}}}\cdot C_{\kappa }^{(\alpha )}(X),}

where  κ ⊢ k {\displaystyle \kappa \vdash k} means  κ {\displaystyle \kappa } is a partition of  k {\displaystyle k} ,  ( a i ) κ ( α ) {\displaystyle (a_{i})_{\kappa }^{(\alpha )}} is the generalized Pochhammer symbol, and  C κ ( α ) ( X ) {\displaystyle C_{\kappa }^{(\alpha )}(X)} is the "C" normalization of the Jack function.

## Two matrix arguments

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If  X {\displaystyle X} and  Y {\displaystyle Y} are two  m × m {\displaystyle m\times m} complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:

p F q ( α ) ( a 1 , … , a p ; b 1 , … , b q ; X , Y ) = ∑ k = 0 ∞ ∑ κ ⊢ k 1 k ! ⋅ ( a 1 ) κ ( α ) ⋯ ( a p ) κ ( α ) ( b 1 ) κ ( α ) ⋯ ( b q ) κ ( α ) ⋅ C κ ( α ) ( X ) C κ ( α ) ( Y ) C κ ( α ) ( I ) , {\displaystyle _{p}F_{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})_{\kappa }^{(\alpha )}\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)}},}

where  I {\displaystyle I} is the identity matrix of size  m {\displaystyle m} .

## Not a typical function of a matrix argument

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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

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

The thing to remember is that

α = 2 β . {\displaystyle \alpha ={\frac {2}{\beta }}.}

Care should be exercised as to whether a particular text is using a parameter  α {\displaystyle \alpha } or  β {\displaystyle \beta } and which the particular value of that parameter is.

Typically, in settings involving real random matrices,  α = 2 {\displaystyle \alpha =2} and thus  β = 1 {\displaystyle \beta =1} . In settings involving complex random matrices, one has  α = 1 {\displaystyle \alpha =1} and  β = 2 {\displaystyle \beta =2} .

## References

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* 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

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

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