On a generalization of the Rogers generating function
Copyright policy )On a generalization of the Rogers generating function
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 ${}_2��_1$ and ${}_8��_7$.
arXiv admin note: text overlap with arXiv:1411.1371
- National Institute of Standards and Technology United States
- University of Maryland United States
- University of Maryland, College Park United States
- Department of Mathematics University of Maryland United States
- University of Maryland, College Park United States
basic hypergeometric series, definite integrals, connection coefficients, 33C45, 05A15, 33C20, 34L10, 30E20, basic hypergeometric orthogonal polynomials, \(q\)-gamma functions, \(q\)-beta functions and integrals, Mathematics - Classical Analysis and ODEs, generating functions, Classical Analysis and ODEs (math.CA), FOS: Mathematics, eigenfunction expansions
basic hypergeometric series, definite integrals, connection coefficients, 33C45, 05A15, 33C20, 34L10, 30E20, basic hypergeometric orthogonal polynomials, \(q\)-gamma functions, \(q\)-beta functions and integrals, Mathematics - Classical Analysis and ODEs, generating functions, Classical Analysis and ODEs (math.CA), FOS: Mathematics, eigenfunction expansions
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