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Two operator identities and their applications to terminating basic hypergeometric series and \(q\)-integrals. (English) Zbl 1081.33032

In this paper the authors prove first two identities involving certain \(q\)-operators and the basic hypergeometric series \(_{3}\phi_{2}\). These are extensions of some earlier results due to W. Y. C. Chen and Z. G. Liu, [Prog. Math. 161, 111–129 (1998; Zbl 0901.33008) and J. Comb. Theory, Ser. A 80, No. 2, 175–195 (1997; Zbl 0901.33009)]. Then applying these operator identities they obtain some summation formulae for terminating basic hypergeometric series. They also obtain some new identities for \(q\)-integrals. These extend some known identities for \(q\)-integrals of G. Gasper [Topics in Mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 294–314 (1989; Zbl 0748.33012)] and M. E. H. Ismail, D. Stanton and G. Viennot [Eur. J. Comb. 8, 379–392 (1987; Zbl 0642.33006)]. All the formulae derived in this paper involve the function \(_{3}\phi_{2}\).

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
05A30 \(q\)-calculus and related topics
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