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On lifting q-difference operators for a chain of basic hypergeometric polynomials

Commun. Math. Anal. Conf.
Conference 3 (2011), 1 - 22

On lifting q-difference operators for a chain of basic hypergeometric polynomials

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Abstract

We construct lifting q-difference operators for a chain of basic hypergeometric polynomials, which composed of the continuous q-Hermite polynomials Hn(x jq) of Rogers, the continuous big q-Hermite polynomials Hn(x;ajq), the Al-Salam–Chihara polynomials Qn(x;a;bjq), the continuous dual q-Hahn polynomials pn(x;a;b;c jq) and, finally, the Askey–Wilson polynomials pn(x;a;b;c;d jq) on the five different levels within the Askey q-scheme. At the first step from the continuous q-Hermite polynomials Hn(x jq) to the continuous big q-Hermite polynomials Hn(x;ajq), the required one-parameter lifting operator is defined as Exton’s q-exponential function eq(aqDq) in terms of the Askey–Wilson divided q-difference operator Dq and it represents a particular q-extension of the standard shift operator exp ³ a d dx ´ . We show next that one can move three steps more upwards in order to reach all other polynomial families in this chain. In all those cases lifting operators turn out to be convolution-type products of two, three and four, respectively, one-parameter q-difference operators of the same type eq(aqDq) at the initial step. At each step, we also determine lifting q-difference operators that enable one to express all the orthogonality weight functions for Hn(x;ajq), Qn(x;a;bjq), pn(x;a;b;cjq) and pn(x;a;b;c;d jq), in terms of the single weight function for the continuous q-Hermite polynomials Hn(x jq) of Rogers.