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1: 35.1 Special Notation
β–Ί β–Ίβ–Ίβ–Ίβ–Ί
a , b complex variables.
𝟎 zero matrix.
𝐈 identity matrix.
β–ΊThe main functions treated in this chapter are the multivariate gamma and beta functions, respectively Ξ“ m ⁑ ( a ) and B m ⁑ ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A Ξ½ ⁑ ( 𝐓 ) and (of the second kind) B Ξ½ ⁑ ( 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) Ξ¨ ⁑ ( 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 ) . … β–ΊRelated notations for the Bessel functions are π’₯ Ξ½ + 1 2 ⁒ ( m + 1 ) ⁑ ( 𝐓 ) = A Ξ½ ⁑ ( 𝐓 ) / A Ξ½ ⁑ ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ⁑ ( 0 , , 0 , Ξ½ | 𝐒 , 𝐓 ) = | 𝐓 | Ξ½ ⁒ B Ξ½ ⁑ ( 𝐒 ⁒ 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 Ξ½ ⁑ ( 𝐓 ) = | 𝐓 | Ξ½ ⁒ B Ξ½ ⁑ ( 𝐒 ⁒ 𝐓 ) (Faraut and Korányi (1994, pp. 357–358)).
2: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
β–Ί
§35.5(i) Definitions
β–Ί
§35.5(ii) Properties
β–Ί
§35.5(iii) Asymptotic Approximations
β–ΊFor asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
3: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
β–Ί
§35.6(i) Definitions
β–Ί
Laguerre Form
β–Ί
§35.6(ii) Properties
β–Ί
§35.6(iii) Relations to Bessel Functions of Matrix Argument
4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
β–Ί
§35.8(i) Definition
β–Ί
Convergence Properties
β–Ί
Confluence
β–Ί
Invariance
5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
β–Ί
§35.7(i) Definition
β–Ί
Jacobi Form
β–Ί
Confluent Form
β–Ί
Integral Representation
6: 35.10 Methods of Computation
§35.10 Methods of Computation
β–ΊSee 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). …
7: 35.9 Applications
§35.9 Applications
β–ΊIn 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. … β–ΊThese references all use results related to the integral formulas (35.4.7) and (35.5.8). … β–ΊIn 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. …
8: 35 Functions of Matrix Argument
Chapter 35 Functions of Matrix Argument
9: 35.2 Laplace Transform
§35.2 Laplace Transform
β–Ί
Definition
β–ΊFor any complex symmetric matrix 𝐙 , … β–Ί
Inversion Formula
β–Ί
Convolution Theorem
10: 1.1 Special Notation
β–Ί β–Ίβ–Ίβ–Ίβ–Ίβ–Ίβ–Ί
x , y real variables.
𝐀 1 inverse of the square matrix 𝐀
𝐈 identity matrix
det ( 𝐀 ) determinant of the square matrix 𝐀
tr ⁑ ( 𝐀 ) trace of the square matrix 𝐀
β–ΊIn the physics, applied maths, and engineering literature a common alternative to a ¯ is a , a being a complex number or a matrix; the Hermitian conjugate of 𝐀 is usually being denoted 𝐀 .