FoCM 2014 conference

Workshop C5 - Special Functions and Orthogonal Polynomials - Semi-plenary talk

December 20, 17:00 ~ 17:50 - Room A21

## Lifting $q$-difference operators in the Askey scheme of basic hypergeometric polynomials

### Natig Atakishiyev

### Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, Mexico - natig@matcuer.unam.mx

We construct an explicit form of a $q$-difference operator that lifts the continuous $q$-Hermite polynomials $H_n(x|q)$ of Rogers into the Askey-Wilson polynomials $p_n(x;a,b,c,d|q)$ on the top level in the Askey $q$-scheme. This operator represents a special convolution-type product of four one-parameter $q$-difference operators of the form $\epsilon_q(c_q D_q)$, defined as Exton's $q$-exponential function $\epsilon_q(z)$ in terms of the Askey-Wilson divided $q$-difference operator $D_q$. We show also that one can determine another $q$-difference operator that transforms the orthogonality weight function for the continuous $q$-Hermite polynomials $H_n(x|q)$ of Rogers up to the weight function, associated with the Askey-Wilson polynomials $p_n(x;a,b,c,d|q)$.

Joint work with Mesuma Atakishiyeva (Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Cuernavaca, Morelos, México).