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

Rahman, Mizan

doi:10.1090/S0002-9939-1984-0759666-X

A simple evaluation of Askey and Wilson’s $q$-beta integral
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Proc. Amer. Math. Soc. 92 (1984), 413-417 Request permission

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 \[ \int _{ - 1}^1 {\frac {{h(x;1)h(x; - 1)h(x;\sqrt q )h(x: - \sqrt {q)} }}{{h(x;a)h(x;b)h(x;c)h(x;d)}}} \frac {{dx}}{{\sqrt {1 - {x^2}} }},\] where $\max \left ( {\left | q \right |,\left | a \right |,\left | b \right |,\left | c \right |,\left | d \right | < 1} \right )$.

References

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