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q-Fractional Askey-Wilson Integrals and Related Semigroups of Operators

Research paper by Mourad E. H. Ismail, Ruiming Zhang, Keru Zhou

Indexed on: 16 Dec '20Published on: 10 Dec '20Published in: arXiv - Mathematics - Classical Analysis and ODEs



Abstract

We introduce three one-parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey-Wilson operator. We also study these families as families of positive linear approximation operators. Applications include connection relations and bilinear formulas for the Askey-Wilson polynomials. We also introduce a q-Gauss-Weierstrass transform and prove a representation and inversion theorem for it.