Two operator identities and their applications to terminating basic hypergeometric series and \(q\)-integrals (Q2575047)

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 \textit{W. Y. C. Chen} and \textit{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 \textit{G. Gasper} [Topics in Mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 294--314 (1989; Zbl 0748.33012)] and \textit{M. E. H. Ismail, D. Stanton} and \textit{G. Viennot} [Eur. J. Comb. 8, 379--392 (1987; Zbl 0642.33006)]. All the formulae derived in this paper involve the function \(_{3}\phi_{2}\).