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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 of one variable, ${}_{r}{\phi }_{s}$
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 · doi:10.2307/2043995 [2] Andrews, G. E.: Q-series: their development and applications in analysis, number theory, combinatorics, physics and computer algebra, CBMS regional conference lecture series 66 (1986) · 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) [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 · doi:10.1016/S0012-365X(02)00626-X [7] Sears, D. B.: Transformation of basic hypergeometric functions of special type, Proc. London math. Soc. 52, 467-483 (1951) · Zbl 0042.07503 · doi:10.1112/plms/s2-52.6.467