A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point
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 $α<0$, outside the support $\{0,1,\dots,N\}$ of the binomial measure, acts through the $j$-th forward difference $\dif^{j}$ (here $\Nzero=\{0,1,2,\dots\}$, $\Z_{+}=\{1,2,\dots\}$, $\R_{+}=(0,\infty)$). In the joint scaling $n,N\to\infty$, $n/N\to r$ with $0 0$ and $j\ge1$; 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 $λ$ concerns each fixed $λ>0$, not the continuity at $λ=0$. Multiprecision computations illustrate the convergence and the exact value $C_{\ast}=2.8$ for $p=0.3$, $r=3/5$, $α=-2.5$, $j=2$, $λ=5$.