A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point

Abstract

We study the monic polynomials $\Sob{n}$ orthogonal with respect to a Sobolev-type modification of the discrete Krawtchouk inner product in which a single point mass at an exterior point $\alpha <0$, outside the support $\{0,1,\dots ,N\}$ of the binomial measure, acts through the $j$-th forward difference $\dif^{j}$ (here $\Nzero=\{0,1,2,\dots\}$, ${\mathbb{Z}}_{+}=\{1,2,\dots \}$, ${\mathbb{R}}_{+}=(0,\infty )$). In the joint scaling $n,N\to \infty$, $n/N\to r$ with $0<p<r<1$ (the \emph{reciprocal-Gamma} regime of Dominici's classification), we prove the Mehler--Heine formula. The exterior mass survives the large-degree limit exactly through the linear factor $z-\alpha$, with the explicit universal constant $C_{\ast }$, and the limit is independent of both $\lambda >0$ and $j\ge 1$; the same conclusion holds in the Uvarov case $j=0$ (Corollary~\ref{cor:uvarov}). The derivation is pointwise and self-contained: it uses only a rank-one connection formula (Theorem~\ref{thm:connection}), the reproducing-kernel recursion, Dominici's pointwise limit and the classical three-term recurrence; the normalised Sobolev correction obeys an exact affine recursion at fixed $N$, controlled by a triangular contraction argument over two backward windows with an explicit boundary estimate, and the contraction ratio $\varrho <1$ is exactly the condition $r>p$. We also discuss the scale of the theorem: in this normalisation the classical term is asymptotically negligible, and the independence of $\lambda$ concerns each fixed $\lambda >0$, not the continuity at $\lambda =0$. Multiprecision computations illustrate the convergence and the exact value $C_{\ast }=2.8$ for $p=0.3$, $r=3/5$, $\alpha =-2.5$, $j=2$, $\lambda =5$.