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

We construct a q-difference operator that lifts the continuous q-Hermite polynomials H-n(x vertical bar q) of Rogers up to the continuous big q-Hermite polynomials H-n(x; a vertical bar q) on the next level in the Askey scheme of basic hypergeometric polynomials. This operator is defined as Exton's q-exponential function epsilon(q)(a(q)D(q)) in terms of the Askey-Wilson divided q-difference operator D-q and it represents a particular q-extension of the standard shift operator exp(ad/dx). We next show that one can move two steps more upwards in order first to reach the Al-Salam-Chihara family of polynomials Q(n)(x; a, b vertical bar q), and then the continuous dual q-Hahn polynomials p(n)(x; a, b, c vertical bar 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 epsilon(q)(a(q)D(q)) at the initial step. At each step, we also determine q-difference operators that lift the weight function for the continuous q-Hermite polynomials H-n(x vertical bar q) successively up to the weight functions for H-n(x; a vertical bar q), Q(n)(x; a, b vertical bar q) and p(n)(x; a, b, c vertical bar q).