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1: 35.1 Special Notation
a , b complex variables.
f ( 𝐗 ) complex-valued function with 𝐗 𝛀 .
The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( 𝐓 ) and (of the second kind) B ν ( 𝐓 ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; 𝐓 ) or F 1 1 ( a b ; 𝐓 ) and (of the second kind) Ψ ( a ; b ; 𝐓 ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; 𝐓 ) or F 1 2 ( a 1 , a 2 b ; 𝐓 ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; 𝐓 ) or F q p ( a 1 , , a p b 1 , , b q ; 𝐓 ) . … Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( 𝐓 ) = A ν ( 𝐓 ) / A ν ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | 𝐒 , 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( 𝐓 ) = | 𝐓 | ν B ν ( 𝐒 𝐓 ) (Faraut and Korányi (1994, pp. 357–358)).
2: 35.5 Bessel Functions of Matrix Argument
§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).
3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(i) Definition
Kummer Transformation
Pfaff–Saalschütz Formula
Thomae Transformation
4: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7(i) Definition
Jacobi Form
Confluent Form
Integral Representation
5: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6(i) Definitions
Laguerre Form
§35.6(ii) Properties
§35.6(iii) Relations to Bessel Functions of Matrix Argument
6: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by … and the summation is over all nonnegative integers s such that the arguments in the factorials are nonnegative. … For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
7: 11.8 Analogs to Kelvin Functions
§11.8 Analogs to Kelvin Functions
For properties of Struve functions of argument x e ± 3 π i / 4 see McLachlan and Meyers (1936).
8: 35.9 Applications
§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 F q p , with p 2 and q 1 . 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. …
9: 35.10 Methods of Computation
§35.10 Methods of Computation
See Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on 𝐎 ( m ) applied to a generalization of the integral (35.5.8). …
10: 35 Functions of Matrix Argument
Chapter 35 Functions of Matrix Argument