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1: 5.18 q -Gamma and q -Beta Functions
§5.18 q -Gamma and q -Beta Functions
§5.18(iii) q -Beta Function
5.18.11 B q ( a , b ) = Γ q ( a ) Γ q ( b ) Γ q ( a + b ) .
5.18.12 B q ( a , b ) = 0 1 t a - 1 ( t q ; q ) ( t q b ; q ) d q t , 0 < q < 1 , a > 0 , b > 0 .
2: 5.21 Methods of Computation
For the computation of the q -gamma and q -beta functions see Gabutti and Allasia (2008).
3: Bibliography I
  • M. E. H. Ismail and D. R. Masson (1994) q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Amer. Math. Soc. 346 (1), pp. 63–116.
  • 4: 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.
  • 5: Bibliography P
  • E. Pairman (1919) Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),
  • R. B. Paris (2002c) Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.
  • 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.
  • H. N. Phien (1990) A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.
  • 6: 18.28 Askey–Wilson Class
    18.28.13 C n ( cos θ ; β | q ) = = 0 n ( β ; q ) ( β ; q ) n - ( q ; q ) ( q ; q ) n - e i ( n - 2 ) θ = ( β ; q ) n ( q ; q ) n e i n θ ϕ 1 2 ( q - n , β β - 1 q 1 - n ; q , β - 1 q e - 2 i θ ) .
    18.28.15 1 2 π 0 π C n ( cos θ ; β | q ) C m ( cos θ ; β | q ) | ( e 2 i θ ; q ) ( β e 2 i θ ; q ) | 2 d θ = ( β , β q ; q ) ( β 2 , q ; q ) ( 1 - β ) ( β 2 ; q ) n ( 1 - β q n ) ( q ; q ) n δ n , m , - 1 < β < 1 .
    18.28.19 R n ( x ) = R n ( x ; α , β , γ , δ | q ) = = 0 n q ( q - n , α β q n + 1 ; q ) ( α q , β δ q , γ q , q ; q ) j = 0 - 1 ( 1 - q j x + γ δ q 2 j + 1 ) = ϕ 3 4 ( q - n , α β q n + 1 , q - y , γ δ q y + 1 α q , β δ q , γ q ; q , q ) , α q , β δ q , or γ q = q - N ; n = 0 , 1 , , N .
    7: Errata
  • Section 17.2(i)

    A sentence was added recommending §27.14(ii) for properties of ( q ; q ) .

  • Equation (18.27.6)

    18.27.6 P n ( α , β ) ( x ; c , d ; q ) = c n q - ( α + 1 ) n ( q α + 1 , - q α + 1 c - 1 d ; q ) n ( q , - q ; q ) n P n ( q α + 1 c - 1 x ; q α , q β , - q α c - 1 d ; q )

    Originally the first argument to the big q -Jacobi polynomial on the right-hand side was written incorrectly as q α + 1 c - 1 d x .

    Reported 2017-09-27 by Tom Koornwinder.

  • Section 17.9

    The title was changed from Transformations of Higher ϕ r r Functions to Further Transformations of ϕ r r + 1 Functions.

  • Section 17.1

    The notation used for the q -Appell functions in Equations (17.4.5), (17.4.6),(17.4.7), (17.4.8), (17.11.1), (17.11.2) and (17.11.3) was updated to explicitly include the argument q , as used in Gasper and Rahman (2004).

  • Equation (17.13.3)
    17.13.3 0 t α - 1 ( - t q α + β ; q ) ( - t ; q ) d t = Γ ( α ) Γ ( 1 - α ) Γ q ( β ) Γ q ( 1 - α ) Γ q ( α + β )

    Originally the differential was identified incorrectly as d q t ; the correct differential is d t .

    Reported 2011-04-08.

  • 8: 18.27 q -Hahn Class
    §18.27(ii) q -Hahn Polynomials
    §18.27(iii) Big q -Jacobi Polynomials
    §18.27(iv) Little q -Jacobi Polynomials
    §18.27(v) q -Laguerre Polynomials
    Discrete q -Hermite II