Mehler--Heine asymptotics for finite differences of mass-modified Charlier and Meixner polynomials

Abstract

Point-mass perturbations and finite-difference operations alter the finite-degree structure of discrete orthogonal polynomials, but their combined effect at the Mehler--Heine scale is not immediate. We study the monic Charlier and Meixner families under a pure Uvarov modification by a fixed mass $A\delta_{0}$ at the endpoint of the support and determine the asymptotic behaviour of their forward and backward differences of arbitrary fixed order. Using a rank-one connection formula, we show that the perturbation coefficient decays factorially in the Charlier case and exponentially, with an algebraic prefactor, in the Meixner case. In both families this decay is faster than every algebraic power of $n^{-1}$ and therefore suppresses the growth produced by any fixed number of finite differences. Consequently, for every fixed $A\geq0$ and $k\in\Nzero$, the mass-modified and classical families have the same locally uniform Mehler--Heine limits in $\C$. Forward differences preserve the reciprocal-Gamma profile up to the factor $(-1)^k$, whereas backward differences translate the limiting argument by $k$, shifting the limiting zero lattice from $\Nzero$ to $k+\Nzero$. We also derive a first-order shift equation for the common forward limit and characterize its entire solution space. Complex-plane portraits and real-axis computations illustrate the predicted limiting profiles, the two zero lattices, and the asymptotic disappearance of the fixed endpoint mass. These results identify a scale-separation mechanism governing the fixed-order asymptotic stability of the Charlier and Meixner families under rank-one endpoint perturbations.