## 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

### Keywords:

difference operator; Nassrallah-Rahman integral

Zbl 0901.33008
Full Text:

### References:

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