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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A simple evaluation of Askey and Wilson's $ q$-beta integral


Author: Mizan Rahman
Journal: Proc. Amer. Math. Soc. 92 (1984), 413-417
MSC: Primary 33A15; Secondary 05A30
MathSciNet review: 759666
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Abstract: By using the well-known sum of $ _2{\phi _1}\left( {a,b;c;c/ab} \right)$ and Sears' identity for the sum of two nonterminating balanced $ _3{\phi _2}$ series, a simple evaluation is given for Askey and Wilson's $ q$-beta type integral

$\displaystyle \int_{ - 1}^1 {\frac{{h(x;1)h(x; - 1)h(x;\sqrt q )h(x: - \sqrt {q)} }}{{h(x;a)h(x;b)h(x;c)h(x;d)}}} \frac{{dx}}{{\sqrt {1 - {x^2}} }},$

where $ \max \left( {\left\vert q \right\vert,\left\vert a \right\vert,\left\vert b \right\vert,\left\vert c \right\vert,\left\vert d \right\vert < 1} \right)$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0759666-X
PII: S 0002-9939(1984)0759666-X
Keywords: $ q$-beta integral, $ q$-binomial theorems, Sears' identity, Askey and Wilson's integral
Article copyright: © Copyright 1984 American Mathematical Society