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1: 35.3 Multivariate Gamma and Beta Functions
§35.3 Multivariate Gamma and Beta Functions
35.3.3 B m ( a , b ) = 0 < X < I | X | a - 1 2 ( m + 1 ) | I - X | b - 1 2 ( m + 1 ) d X , ( a ) , ( b ) > 1 2 ( m - 1 ) .
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 ) . … 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
35.6.6 B m ( b 1 , b 2 ) | T | b 1 + b 2 - 1 2 ( m + 1 ) F 1 1 ( a 1 + a 2 b 1 + b 2 ; T ) = 0 < X < T | X | b 1 - 1 2 ( m + 1 ) F 1 1 ( a 1 b 1 ; X ) | T - X | b 2 - 1 2 ( m + 1 ) F 1 1 ( a 2 b 2 ; T - X ) d X , ( b 1 ) , ( b 2 ) > 1 2 ( m - 1 ) .
4: 35.4 Partitions and Zonal Polynomials
5: 19.16 Definitions
19.16.9 R - a ( b ; z ) = 1 B ( a , a ) 0 t a - 1 j = 1 n ( t + z j ) - b j d t = 1 B ( a , a ) 0 t a - 1 j = 1 n ( 1 + t z j ) - b j d t , b 1 + + b n > a > 0 , b j , z j ( - , 0 ] ,
19.16.12 R - a ( b 1 , , b 4 ; c - 1 , c - k 2 , c , c - α 2 ) = 2 ( sin 2 ϕ ) 1 - a B ( a , a ) 0 ϕ ( sin θ ) 2 a - 1 ( sin 2 ϕ - sin 2 θ ) a - 1 ( cos θ ) 1 - 2 b 1 ( 1 - k 2 sin 2 θ ) - b 2 ( 1 - α 2 sin 2 θ ) - b 4 d θ ,
19.16.19 R - a ( b 1 , , b n ; 0 , z 2 , , z n ) = B ( a , a - b 1 ) B ( a , a ) R - a ( b 2 , , b n ; z 2 , , z n ) , a + a > 0 , a > b 1 .
19.16.24 R - a ( b ; z ) = z 1 a - b 1 B ( b 1 , a - b 1 ) 0 t b 1 - 1 ( t + z 1 ) - a R - a ( b ; 0 , t + z 2 , , t + z n ) d t , a > b 1 , a + a > b 1 > 0 .
6: 19.23 Integral Representations
19.23.8 R - a ( b ; z ) = 2 B ( b 1 , b 2 ) 0 π / 2 ( z 1 cos 2 θ + z 2 sin 2 θ ) - a ( cos θ ) 2 b 1 - 1 ( sin θ ) 2 b 2 - 1 d θ , b 1 , b 2 > 0 ; z 1 , z 2 > 0 .
19.23.10 R - a ( b ; z ) = 1 B ( a , a ) 0 1 u a - 1 ( 1 - u ) a - 1 j = 1 n ( 1 - u + u z j ) - b j d u , a , a > 0 ; a + a = j = 1 n b j ; z j ( - , 0 ] .
7: 35.7 Gaussian Hypergeometric Function of Matrix Argument
8: 35.8 Generalized Hypergeometric Functions of Matrix Argument
35.8.13 0 < X < I | X | a 1 - 1 2 ( m + 1 ) | I - X | b 1 - a 1 - 1 2 ( m + 1 ) F q p ( a 2 , , a p + 1 b 2 , , b q + 1 ; T X ) d X = 1 B m ( b 1 - a 1 , a 1 ) F q + 1 p + 1 ( a 1 , , a p + 1 b 1 , , b q + 1 ; T ) , ( b 1 - a 1 ) , ( a 1 ) > 1 2 ( m - 1 ) .
9: 19.28 Integrals of Elliptic Integrals
Also, B again denotes the beta function5.12). …
19.28.1 0 1 t σ - 1 R F ( 0 , t , 1 ) d t = 1 2 ( B ( σ , 1 2 ) ) 2 ,
19.28.2 0 1 t σ - 1 R G ( 0 , t , 1 ) d t = σ 4 σ + 2 ( B ( σ , 1 2 ) ) 2 ,
19.28.3 0 1 t σ - 1 ( 1 - t ) R D ( 0 , t , 1 ) d t = 3 4 σ + 2 ( B ( σ , 1 2 ) ) 2 .
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 ) .
10: Bibliography P
  • E. Pairman (1919) Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),
  • P. I. Pastro (1985) Orthogonal polynomials and some q -beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
  • K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.
  • M. D. Perlman and I. Olkin (1980) Unbiasedness of invariant tests for MANOVA and other multivariate problems. Ann. Statist. 8 (6), pp. 1326–1341.
  • H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.