A simple evaluation of Askey and Wilson’s $q$-beta integral
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- by Mizan Rahman PDF
- 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
- W. A. Al-Salam and A. Verma, Some remarks on $q$-beta integral, Proc. Amer. Math. Soc. 85 (1982), no. 3, 360–362. MR 656102, DOI 10.1090/S0002-9939-1982-0656102-7
- 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 10.1090/S0002-9939-1981-0589145-1
- Richard Askey, The $q$-gamma and $q$-beta functions, Applicable Anal. 8 (1978/79), no. 2, 125–141. MR 523950, DOI 10.1080/00036817808839221
- Richard Askey, An elementary evaluation of a beta type integral, Indian J. Pure Appl. Math. 14 (1983), no. 7, 892–895. MR 714840
- Richard Askey and James Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55. MR 783216, DOI 10.1090/memo/0319
- D. B. Sears, Transformations of basic hypergeometric functions of special type, Proc. London Math. Soc. (2) 52 (1951), 467–483. MR 41982, DOI 10.1112/plms/s2-52.6.467
- Lucy Joan Slater, Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966. MR 0201688 J. Thomae, Beiträge zur Theorie der durch die Heinesche Reihe: $1 + \left ( {\left ( {1 - {q^\alpha }} \right )\left ( {1 - {q^\beta }} \right )/\left ( {1 - q} \right )\left ( {1 - {q^\gamma }} \right )} \right )x + \cdots$ darstellbaren Functionen, J. Reine Angew. Math. 70 (1869), 258-281.
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 413-417
- MSC: Primary 33A15; Secondary 05A30
- DOI: https://doi.org/10.1090/S0002-9939-1984-0759666-X
- MathSciNet review: 759666