TY - JOUR
T1 - On a generalization of the Rogers generating function
AU - Cohl, Howard S.
AU - Costas-Santos, Roberto S.
AU - Wakhare, Tanay V.
JO - Journal of Mathematical Analysis and Applications
VL - 475
IS - 2
SP - 1019
EP - 1043
PY - 2019
DA - 2019/07/15/
SN - 0022-247X
DO - https://doi.org/10.1016/j.jmaa.2019.01.068
UR - https://www.sciencedirect.com/science/article/pii/S0022247X19301027
KW - Basic hypergeometric series
KW - Basic hypergeometric orthogonal polynomials
KW - Generating functions
KW - Connection coefficients
KW - Eigenfunction expansions
KW - Definite integrals
AB - We derive a generalization of the Rogers generating function for the continuous q-ultraspherical/Rogers polynomials whose coefficient is a ϕ12. 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 ϕ78, 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 ϕ78 to a ϕ12. We also obtain several definite integral representations which correspond to the above mentioned expansions through the use of orthogonality.
ER -