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A remark on Andrews-Askey integral. (English) Zbl 1142.33006

Summary: We use the Andrews-Askey integral and the \(q\)-Chu-Vandermonde formula to derive a more general integral formula. Applications of the new integral formula are also given, which include to derive the \(q\)-Pfaff-Saalschütz formula and the terminating Sears’s \(_3\phi _2\) transformation formula.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
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References:

[1] Andrews, G. E.; Askey, R., Another \(q\)-extension of the beta function, Proc. Amer. Math. Soc., 81, 97-100 (1981) · Zbl 0471.33001
[2] Andrews, G. E., \(q\)-Series: Their Development and Applications in Analysis, Number Theory, Combinatorics, Physics and Computer Algebra, CBMS Regional Conference Lecture Series, vol. 66 (1986), Amer. Math. Soc.: Amer. Math. Soc. Providences, RI · Zbl 0594.33001
[3] Carlitz, L., A \(q\)-identity, Fibonacci Quart., 12, 369-372 (1974) · Zbl 0296.33001
[4] Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, MA · Zbl 0695.33001
[5] Jackson, F. H., On \(q\)-definite integrals, Q. J. Pure Appl. Math., 50, 101-112 (1910)
[6] Liu, Z.-G., Some operator identities and \(q\)-series transformation formulas, Discrete Math., 265, 119-139 (2003) · Zbl 1021.05010
[7] Sears, D. B., Transformation of basic hypergeometric functions of special type, Proc. London Math. Soc., 52, 467-483 (1951) · Zbl 0042.07503
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