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On a generalization of the Rogers generating function. (English) Zbl 1461.33007
Summary: We derive a generalization of the Rogers generating function for the continuous \(q\)-ultraspherical/Rogers polynomials whose coefficient is a \(_2\phi_1\). From that expansion, we derive corresponding specialization and limit transition expansions for the continuous \(q\)-Hermite, continuous \(q\)-Legendre, Laguerre, and Chebyshev polynomials of the first kind. Using a generalized expansion of the Rogers generating function in terms of the Askey-Wilson polynomials by Ismail & Simeonov whose coefficient is a \(_8\phi_7\), we derive corresponding generalized expansions for the Wilson, continuous \(q\)-Jacobi, and Jacobi polynomials. By comparing the coefficients of the Askey-Wilson expansion to our continuous \(q\)-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric functions which relates an \(_8\phi_7\) to a \(_2\phi_1\). We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality.

MSC:
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
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