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11: Bibliography P
  • P. I. Pastro (1985) Orthogonal polynomials and some q -beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.
  • 12: 8.24 Physical Applications
    §8.24 Physical Applications
    13: 8.1 Special Notation
    Unless otherwise indicated, primes denote derivatives with respect to the argument. The functions treated in this chapter are the incomplete gamma functions γ ( a , z ) , Γ ( a , z ) , γ * ( a , z ) , P ( a , z ) , and Q ( a , z ) ; the incomplete beta functions B x ( a , b ) and I x ( a , b ) ; the generalized exponential integral E p ( z ) ; the generalized sine and cosine integrals si ( a , z ) , ci ( a , z ) , Si ( a , z ) , and Ci ( a , z ) . Alternative notations include: Prym’s functions P z ( a ) = γ ( a , z ) , Q z ( a ) = Γ ( a , z ) , Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); ( a , z ) ! = γ ( a + 1 , z ) , [ a , z ] ! = Γ ( a + 1 , z ) , Dingle (1973); B ( a , b , x ) = B x ( a , b ) , I ( a , b , x ) = I x ( a , b ) , Magnus et al. (1966); Si ( a , x ) Si ( 1 - a , x ) , Ci ( a , x ) Ci ( 1 - a , x ) , Luke (1975).
    14: 19.26 Addition Theorems
    19.26.13 R C ( α 2 , α 2 - θ ) + R C ( β 2 , β 2 - θ ) = R C ( σ 2 , σ 2 - θ ) , σ = ( α β + θ ) / ( α + β ) ,
    19.26.17 α R C ( β , α + β ) + β R C ( α , α + β ) = π / 2 , α , β ( - , 0 ) , α + β > 0 .
    19.26.22 R J ( x , y , z , p ) = 2 R J ( x + λ , y + λ , z + λ , p + λ ) + 3 R C ( α 2 , β 2 ) ,
    15: 8.17 Incomplete Beta Functions
    8.17.1 B x ( a , b ) = 0 x t a - 1 ( 1 - t ) b - 1 d t ,
    §8.17(iii) Integral Representation
    8.17.10 I x ( a , b ) = x a ( 1 - x ) b 2 π i c - i c + i s - a ( 1 - s ) - b d s s - x ,
    16: Bibliography
  • W. A. Al-Salam and M. E. H. Ismail (1994) A q -beta integral on the unit circle and some biorthogonal rational functions. Proc. Amer. Math. Soc. 121 (2), pp. 553–561.
  • 17: 35.7 Gaussian Hypergeometric Function of Matrix Argument
    18: 5.20 Physical Applications
    5.20.3 ψ n ( β ) = n e - β W d x = ( 2 π ) n / 2 β - ( n / 2 ) - ( β n ( n - 1 ) / 4 ) ( Γ ( 1 + 1 2 β ) ) - n j = 1 n Γ ( 1 + 1 2 j β ) .
    5.20.5 ψ n ( β ) = 1 ( 2 π ) n [ - π , π ] n e - β W d θ 1 d θ n = Γ ( 1 + 1 2 n β ) ( Γ ( 1 + 1 2 β ) ) - n .
    19: 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 ) .
    20: 16.15 Integral Representations and Integrals
    16.15.1 F 1 ( α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( α ) Γ ( γ - α ) 0 1 u α - 1 ( 1 - u ) γ - α - 1 ( 1 - u x ) β ( 1 - u y ) β d u , α > 0 , ( γ - α ) > 0 ,
    16.15.2 F 2 ( α ; β , β ; γ , γ ; x , y ) = Γ ( γ ) Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β ) Γ ( γ - β ) 0 1 0 1 u β - 1 v β - 1 ( 1 - u ) γ - β - 1 ( 1 - v ) γ - β - 1 ( 1 - u x - v y ) α d u d v , γ > β > 0 , γ > β > 0 ,
    16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
    16.15.4 F 4 ( α , β ; γ , γ ; x ( 1 - y ) , y ( 1 - x ) ) = Γ ( γ ) Γ ( γ ) Γ ( α ) Γ ( β ) Γ ( γ - α ) Γ ( γ - β ) 0 1 0 1 u α - 1 v β - 1 ( 1 - u ) γ - α - 1 ( 1 - v ) γ - β - 1 ( 1 - u x ) γ + γ - α - 1 ( 1 - v y ) γ + γ - β - 1 ( 1 - u x - v y ) α + β - γ - γ + 1 d u d v , γ > α > 0 , γ > β > 0 .