On the q-polynomials in the exponential lattice $x(s)=c_1q +c_3$

Abstract

The main goal of this paper is to continue the study of q-polynomials on non-uniform lattices by using the approach introduced by A. F. Nikiforov and V. B. Uvarov [Akad. Nauk SSSR Inst. Prikl. Mat. Preprint 1983, no. 17, 34 pp.; MR0753537 (86c:39006)]. We consider the q-polynomials on the non-uniform exponential lattice $x(s)=c_1q +c_3$ and study some of their properties (differentiation formulas, structure relations, representation in terms of hypergeometric and basic hypergeometric functions, etc.). Special emphasis is given to q-analogues of the Charlier orthogonal polynomials. For these Charlier polynomials we compute the main data, i.e., the coefficients of the three-term recurrence relation, the structure relation, the square of the norm, etc., in the exponential lattices $x(s)=q $ and $x(s)=(q -1)/(q-1)$, respectively.