Cohl, Howard S. , COSTAS SANTOS, ROBERTO SANTIAGO, Wakhare, Tanay V.
Si
J. Math. Anal. Appl.
Article
Científica
1.22
1.021
15/07/2019
000465168900001
2-s2.0-85062069909
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 807, 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. Published by Elsevier Inc.
Basic hypergeometric series; Basic hypergeometric orthogonal polynomials; Generating functions; Connection coefficients; Eigenfunction expansions; Definite integrals