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1: 35.3 Multivariate Gamma and Beta Functions
§35.3 Multivariate Gamma and Beta Functions
§35.3(i) Definitions
§35.3(ii) Properties
35.3.6 Γ m ( a , , a ) = Γ m ( a ) .
35.3.7 B m ( a , b ) = Γ m ( a ) Γ m ( b ) Γ m ( a + b ) .
2: 35.1 Special Notation
a , b

complex variables.

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 ) . An alternative notation for the multivariate gamma function is Π m ( a ) = Γ m ( a + 1 2 ( m + 1 ) ) (Herz (1955, p. 480)). 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)).
3: 35.6 Confluent Hypergeometric Functions of Matrix Argument
35.6.2 Ψ ( a ; b ; T ) = 1 Γ m ( a ) Ω etr ( - T X ) | X | a - 1 2 ( m + 1 ) | I + X | b - a - 1 2 ( m + 1 ) d X , ( a ) > 1 2 ( m - 1 ) , T Ω .
35.6.3 L ν ( γ ) ( T ) = Γ m ( γ + ν + 1 2 ( m + 1 ) ) Γ m ( γ + 1 2 ( m + 1 ) ) F 1 1 ( - ν γ + 1 2 ( m + 1 ) ; T ) , ( γ ) , ( γ + ν ) > - 1 .
35.6.8 Ω | T | c - 1 2 ( m + 1 ) Ψ ( a ; b ; T ) d T = Γ m ( c ) Γ m ( a - c ) Γ m ( c - b + 1 2 ( m + 1 ) ) Γ m ( a ) Γ m ( a - b + 1 2 ( m + 1 ) ) , ( a ) > ( c ) + 1 2 ( m - 1 ) > m - 1 , ( c - b ) > - 1 .
4: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.5 F 2 3 ( a 1 , a 2 , a 3 b 1 , b 2 ; I ) = Γ m ( b 2 ) Γ m ( c ) Γ m ( b 2 - a 3 ) Γ m ( c + a 3 ) F 2 3 ( b 1 - a 1 , b 1 - a 2 , a 3 b 1 , c + a 3 ; I ) , ( b 2 ) , ( c ) > 1 2 ( m - 1 ) .
35.8.6 F 2 3 ( a 1 , a 2 , a 3 b 1 , b 2 ; I ) = Γ m ( b 1 - a 1 ) Γ m ( b 1 - a 2 ) Γ m ( b 1 ) Γ m ( b 1 - a 1 - a 2 ) Γ m ( b 1 - a 3 ) Γ m ( b 1 - a 1 - a 2 - a 3 ) Γ m ( b 1 - a 1 - a 3 ) Γ m ( b 1 - a 2 - a 3 ) .
35.8.7 F 2 3 ( a 1 , a 2 , a 3 b 1 , b 2 ; I ) = Γ m ( b 1 ) Γ m ( b 2 ) Γ ( c ) Γ m ( a 1 ) Γ m ( c + a 2 ) Γ ( c + a 3 ) F 2 3 ( b 1 - a 1 , b 2 - a 2 , c c + a 2 , c + a 3 ; I ) , ( b 1 ) , ( b 2 ) , ( c ) > 1 2 ( m - 1 ) .
5: 35.4 Partitions and Zonal Polynomials
6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
35.7.2 P ν ( γ , δ ) ( T ) = Γ m ( γ + ν + 1 2 ( m + 1 ) ) Γ m ( γ + 1 2 ( m + 1 ) ) F 1 2 ( - ν , γ + δ + ν + 1 2 ( m + 1 ) γ + 1 2 ( m + 1 ) ; T ) , 0 < T < I ; γ , δ , ν ; ( γ ) > - 1 .
35.7.8 F 1 2 ( a , b c ; T ) = Γ m ( c ) Γ m ( c - a - b ) Γ m ( c - a ) Γ m ( c - b ) F 1 2 ( a , b a + b - c + 1 2 ( m + 1 ) ; I - T ) , 0 < T < I ; 1 2 ( j + 1 ) - a for some j = 1 , , m ; 1 2 ( j + 1 ) - c and c - a - b - 1 2 ( m - j ) for all j = 1 , , m .
7: 19.31 Probability Distributions
19.31.2 n ( x T A x ) μ exp ( - x T B x ) d x 1 d x n = π n / 2 Γ ( μ + 1 2 n ) det B Γ ( 1 2 n ) R μ ( 1 2 , , 1 2 ; λ 1 , , λ n ) , μ > - 1 2 n .
8: 35.5 Bessel Functions of Matrix Argument
9: 19.23 Integral Representations
19.23.9 R - a ( b ; z ) = 4 Γ ( b 1 + b 2 + b 3 ) Γ ( b 1 ) Γ ( b 2 ) Γ ( b 3 ) 0 π / 2 0 π / 2 ( j = 1 3 z j l j 2 ) - a j = 1 3 l j 2 b j - 1 sin θ d θ d ϕ , b j > 0 , z j > 0 .
10: 19.28 Integrals of Elliptic Integrals
19.28.4 0 1 t σ - 1 ( 1 - t ) c - 1 R - a ( b 1 , b 2 ; t , 1 ) d t = Γ ( c ) Γ ( σ ) Γ ( σ + b 2 - a ) Γ ( σ + c - a ) Γ ( σ + b 2 ) , c = b 1 + b 2 > 0 , σ > max ( 0 , a - b 2 ) .