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Some remarks on $ q$-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 | References | Similar Articles | Additional Information

Abstract: The following $ q$-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 $ q$-integral is also considered. Some applications to the $ q$-Wilson (or Askey-Wilson) polynomials are also given.

References [Enhancements On Off] (What's this?)

  • [1] G. E. Andrews and R. Askey, Another $ q$-extension of the beta function, Proc. Amer. Math. Soc. 81 (1981), 97-100. MR 589145 (81j:33001)
  • [2] R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients of $ 6 - j$ 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] A. Dinghas, Zur Darstellung einger Klassen hypergeometrischer Polynome durch Integrale vom Dirichlet-Mehlerschen Typus, Math. Z. 53 (1950), 76-83. MR 0036876 (12:177d)
  • [5] D. B. Sears, Transformation of basic hypergeometric functions of special type, Proc. London Math. Soc. 52 (1951), 467-483. MR 0041982 (13:33e)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0656102-7
Keywords: $ q$-beta integrals, Askey-Wilson polynomials, basic series
Article copyright: © Copyright 1982 American Mathematical Society

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