Inner products involving differences: The Meixner-Sobolev polynomials

Abstract

In this paper, polynomials which are orthogonal with respect to the inner product $$\langle p,q\rangle_S= \sum infty_{s=0} p(s)q(s) {\mu \Gamma (\gamma+s) \over\Gamma(s+1) \Gamma (\gamma)}+ \lambda \sum infty_{s=0} \Delta p(s)\Delta q(s){\mu \Gamma(\gamma+s) \over\Gamma (s+1)\Gamma (\gamma)},$$ where $0<\mu<1$, $\gamma>0$, $\lambda\ge 0$ are studied. For these polynomials, algebraic properties and difference equations are obtained as well as their relation with the Meixner polynomials. Moreover, some properties about the zeros of these polynomials are deduced.