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 \[ \int _{ - c}^d {\frac {{{{( - qt/c)}_{\alpha - 1}}{{(qt/d)}_{\beta - 1}}}} {{{{( - qet)}_{\alpha + \beta }}}}} {d_q}t\] is evaluated. A more general $q$-integral is also considered. Some applications to the $q$-Wilson (or Askey-Wilson) polynomials are also given.
- 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, DOI https://doi.org/10.1090/S0002-9939-1981-0589145-1 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. W. N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, New York, 1935.
- Alexander Dinghas, Zur Darstellung einiger Klassen hypergeometrischer Polynome durch Integrale vom Dirichlet-Mehlerschen Typus, Math. Z. 53 (1950), 76–83 (German). MR 36876, DOI https://doi.org/10.1007/BF01175582
- D. B. Sears, Transformations of basic hypergeometric functions of special type, Proc. London Math. Soc. (2) 52 (1951), 467–483. MR 41982, DOI https://doi.org/10.1112/plms/s2-52.6.467
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Additional Information
Keywords:
<IMG WIDTH="15" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$q$">-beta integrals,
Askey-Wilson polynomials,
basic series
Article copyright:
© Copyright 1982
American Mathematical Society
