…
►
►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively
and
, and the special functions of matrix
argument: Bessel (of the first kind)
and (of the second kind)
; confluent hypergeometric (of the first kind)
or
and (of the second kind)
; Gaussian hypergeometric
or
; generalized hypergeometric
or
.
…
►Related notations for the Bessel functions are
(
Faraut and Korányi (1994, pp. 320–329)),
(
Terras (1988, pp. 49–64)), and
(
Faraut and Korányi (1994, pp. 357–358)).
§35.5 Bessel Functions of Matrix Argument
►
§35.5(i) Definitions
…
►
§35.5(ii) Properties
…
►
§35.5(iii) Asymptotic Approximations
►For asymptotic approximations for Bessel functions of matrix
argument, see
Herz (1955) and
Butler and Wood (2003).
§34.2 Definition: Symbol
►The quantities
in the
symbol are called
angular momenta.
…The corresponding
projective quantum numbers
are given by
…
►and the summation is over all nonnegative integers
such that the
arguments in the factorials are nonnegative.
…
►For alternative expressions for the
symbol, written either as a finite sum or as other terminating generalized hypergeometric series
of unit
argument, see
Varshalovich et al. (1988, §§8.21, 8.24–8.26).
§11.8 Analogs to Kelvin Functions
►For properties of Struve functions of
argument
see
McLachlan and Meyers (1936).
§35.9 Applications
►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix
argument
, with
and
.
See
James (1964),
Muirhead (1982),
Takemura (1984),
Farrell (1985), and
Chikuse (2003) for extensive treatments.
…
►These references all use results related to the integral formulas (
35.4.7) and (
35.5.8).
…
►In chemistry,
Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix
argument, and develop asymptotic approximations for these density functions.
…
§35.10 Methods of Computation
…
►See
Yan (1992) for the
and
functions of matrix
argument in the case
, and
Bingham et al. (1992) for Monte Carlo simulation on
applied to a generalization of the integral (
35.5.8).
…