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On a generalization of the Rogers generating function



Howard Cohl, Roberto Costas-Santos, Tanay Wakhare


We derive a generalized Rogers generating function and corresponding definite integral, for the continuous q-ultraspherical polynomials by applying its connection relation and utilizing orthogonality. Using a recent generalization of the Rogers generating function by Ismail & Simeonov expanded in terms of Askey-Wilson polynomials, we derive corresponding generalized expansions for the continuous q-Jacobi, and Wilson polynomials with two and four free parameters respectively. Comparing the coefficients of the Askey-Wilson expansion to our continuous q-ultraspherical/Rogers expansion, we derive a new quadratic transformation for basic hypergeometric series connecting }_2\phi_1 and }_8\phi_7.
Journal of Mathematical Analysis and Applications


Basic hypergeometric series, Basic hypergeometricorthogonal polynomials, Generating functions, Connectioncoefficients, Eigenfunction expansions, Definite integrals.


Cohl, H. , Costas-Santos, R. and Wakhare, T. (2019), On a generalization of the Rogers generating function, Journal of Mathematical Analysis and Applications, [online],, (Accessed April 18, 2024)
Created July 14, 2019, Updated May 4, 2021