September 30, 2010 Thursday 15:30
Natig Atakishiyev, Instituto de Matematicas, Unidad Cuernavaca Universidad Nacional Autonoma de Mexico
Lifting q-difference operators for a chain of basic hypergeometric polynomials

We construct explicit forms of q-difference operators that lift the continuous q-Hermite polynomials Hn(x| q) of Rogers upwards to reach successively the continuous big q-Hermite polynomials Hn(x; a| q), the Al-Salam–Chihara polynomials Qn(x; a, b | q), the continuous dual q-Hahn polynomials pn(x; a, b, c| q), and, finally, the Askey–Wilson polynomials pn(x; a, b, c, d| q) on the top level in the Askey q-scheme. At the first step the required one-parameter lifting operator is defined as Exton’s q-exponential function "q(aq Dq) in terms of the Askey–Wilson divided q-difference operator Dq and it represents a particular q-extension of the standard shift operator exp (a*d/dx). Lifting operators for q-polynomial families with two, three or four parameters are then determined as convolution-type products of two, three or four, respectively, one-parameter q-difference operators of the same type, as emerges at the first step. At each step we find also q-difference operators that enable one to express all orthogonality weight functions for this chain of q-polynomial families, Hn(x; a| q), Qn(x; a, b | q), p n(x; a, b, c| q) and pn(x; a, b, c, d| q), in terms of the single weight function for the continuous q-Hermite polynomials Hn(x| q) of Rogers.