On lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials

Abstract

We construct a q-difference operator that lifts the continuous q-Hermite polynomials Hn(x| q) of Rogers up to the continuous big q-Hermite polynomials Hn(x; a| q) on the next level in the Askey scheme of basic hypergeometric polynomials. This operator is defined as Exton's q-exponential function εq(aqDq) in terms of the Askey–Wilson divided q-difference operator Dq and it represents a particular q-extension of the standard shift operator . We next show that one can move two steps more upwards in order first to reach the Al-Salam–Chihara family of polynomials Qn(x; a, b | q), and then the continuous dual q-Hahn polynomials pn(x; a, b, c| q). In both these cases, lifting operators, respectively, turn out to be convolution-type products of two and three one-parameter q-difference operators of the same type εq(aqDq) at the initial step. At each step, we also determine q-difference operators that lift the weight function for the continuous q-Hermite polynomials Hn(x| q) successively up to the weight functions for Hn(x; a| q), Qn(x; a, b | q) and pn(x; a, b, c| q).