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

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

Two matrix arguments

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

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

The thing to remember is that

[math]\displaystyle{ \alpha=\frac{2}{\beta}. }[/math]

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

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

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