On generating functions and Mehler--Heine formulas for discrete Charlier and Meixner Sobolev-type orthogonal polynomials

Abstract

Generating functions are among the most important analytical tools in the theory of orthogonal polynomials, providing a unified framework for deriving structural identities, asymptotic expansions, and zero distributions. However, despite the extensive development of discrete Sobolev orthogonal polynomials, no general generating-function theory has been available for the Sobolev-type Charlier and Meixner families associated with arbitrary-order forward differences $j\geq 1$ and an exterior mass point $α<0$. In this paper, we develop the first unified generating-function framework for these families. Starting from explicit connection formulas, we derive generating functions for the Sobolev-type polynomials and their iterated forward differences, which serve as the main analytical tool for establishing new Mehler--Heine formulas. The resulting asymptotic analysis shows that an exterior Sobolev perturbation generates exactly one exceptional zero converging to the mass point, while the remaining zeros preserve the classical asymptotic distribution. Moreover, the limiting Mehler--Heine functions are independent of both the Sobolev mass parameter and the order of the forward difference operator, revealing a universality phenomenon for higher-order discrete Sobolev perturbations. These results considerably extend the analytical theory of discrete Sobolev orthogonal polynomials and establish a direct connection between generating functions, Mehler--Heine asymptotics, and the asymptotic distribution of the zeros for the discrete Sobolev-type Charlier and Meixner families.