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Parameter augmentation for basic hypergeometric series. II. (English) Zbl 0901.33009

[For part I see the authors in Prog. Math. 161, 111–129 (1998; Zbl 0901.33008).]
Let \(D_q\) be the \(q\)-difference operator, \( D_qf(a) = (f(a) - f(aq))/a\), and define an exponential operator \(T\) by \[ T(bD_q) = \sum_{n=0}^{\infty} {(bD_q)^n \over (q;q)_n}. \] The authors derive many known results by applying this operator to simpler results. As an example, they obtain the Askey-Wilson integral by applying this operator three times to the integral identity \[ \int_0^{\pi} {(e^{2i\theta},e^{-2i\theta};q)_{\infty} \over (ae^{i\theta},ae^{-i\theta};q)_{\infty}} d \theta = {2\pi \over (q;q)_{\infty}}. \] A fourth application of their operator yields the Ismail-Stanton-Viennot integral which implies the Nassrallah-Rahman integral. Their technique also yields a particularly simple proof of the linearization formula for the Rogers-Szegő polynomials.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30 \(q\)-calculus and related topics
05A19 Combinatorial identities, bijective combinatorics

Citations:

Zbl 0901.33008
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References:

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