Some remarks on 𝑞-beta integral

Al-Salam, W. A.; Verma, A.

doi:10.1090/S0002-9939-1982-0656102-7

Some remarks on -beta integral

Authors: W. A. Al-Salam and A. Verma
Journal: Proc. Amer. Math. Soc. 85 (1982), 360-362
MSC: Primary 33A15; Secondary 33A65
DOI: https://doi.org/10.1090/S0002-9939-1982-0656102-7
MathSciNet review: 656102
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Abstract: The following -integral

$\displaystyle \int_{ - c}^d {\frac{{{{( - qt/c)}_{\alpha - 1}}{{(qt/d)}_{\beta - 1}}}} {{{{( - qet)}_{\alpha + \beta }}}}} {d_q}t$

is evaluated. A more general

-integral is also considered. Some applications to the

-Wilson (or Askey-Wilson) polynomials are also given.

[1] George E. Andrews and Richard Askey, Another 𝑞-extension of the beta function, Proc. Amer. Math. Soc. 81 (1981), no. 1, 97–100. MR 589145, https://doi.org/10.1090/S0002-9939-1981-0589145-1
[2] R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients of symbols, Mathematics Research Center, University of Wisconsin-Madison, MRC Technical Summary Report #1833.
[3] W. N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, New York, 1935.
[4] Alexander Dinghas, Zur Darstellung einiger Klassen hypergeometrischer Polynome durch Integrale vom Dirichlet-Mehlerschen Typus, Math. Z. 53 (1950), 76–83 (German). MR 36876, https://doi.org/10.1007/BF01175582
[5] D. B. Sears, Transformations of basic hypergeometric functions of special type, Proc. London Math. Soc. (2) 52 (1951), 467–483. MR 41982, https://doi.org/10.1112/plms/s2-52.6.467