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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\ge 0} and be integers, and let be an complex symmetric matrix. Then the hypergeometric function of a matrix argument Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and parameter is defined as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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), }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa\vdash k} means Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa} is a partition of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_i)^{(\alpha )}_{\kappa}} is the generalized Pochhammer symbol, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_\kappa^{(\alpha )}(X)} is the "C" normalization of the Jack function.
Two matrix arguments
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} and are two Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\times m} complex symmetric matrices, then the hypergeometric function of two matrix arguments is defined as:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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)}, }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is the identity matrix of size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} .
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} is omitted. Also, in different publications different values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} are being implicitly assumed. For example, in the theory of real random matrices (see, e.g., Muirhead, 1984), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=2} whereas in other settings (e.g., in the complex case—see Gross and Richards, 1989), . To make matters worse, in random matrix theory researchers tend to prefer a parameter called Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} instead of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} which is used in combinatorics.
The thing to remember is that
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=\frac{2}{\beta}.}
Care should be exercised as to whether a particular text is using a parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} and which the particular value of that parameter is.
Typically, in settings involving real random matrices, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=2} and thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta=1} . In settings involving complex random matrices, one has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta=2} .
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.