Hypergeometric function of a matrix argument: Difference between revisions

Revision as of 23:26, 29 December 2006

Revision as of 06:11, 30 December 2006

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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

<math>C_\kappa^{(\alpha )}(X)</math> is the ``C" normalization of the [[Jack function]].

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==The parameter <math>\alpha</math>==

In many publications the parameter <math>\alpha</math> is omitted. yetAlso, 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 combinatorics researchers often use the parameter <math>\alpha</math> whereas in random matrix theory reserchers tend to prefer a parameter called <math>\beta</math>. Ininstead other disciplines, it isof <math>\alpha/2</math> thatwhich hasis certainused meaningin combinatorics. In either case the connection is simple:

The thing to remember is that

:<math>\alpha=\frac{2}{\beta}</math>

and care:<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>.

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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.

* Kaneko, J. Kaneko, "Selberg Integrals and hypergeometric functions associated with Jack polynomials", ''SIAM Journal on Mathematical Analysis'', '''24''', no. 4, 1086-1110, 1993.

* Koev, Plamen Koev and Edelman, Alan Edelman, "The efficient evaluation of the hypergeometric function of a matrix argument", ''Mathematics of Computation'', '''75''', no. 254, 833-846, 2006.

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