发表机构
Loyola University; Université Quisqueya（洛约拉大学; 克雷斯皮亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文在倒数伽马尺度下证明了带外部质量点的Krawtchouk--Sobolev型正交多项式的Mehler--Heine极限，外部质量通过线性因子存活，极限与参数无关，并给出显式常数。
AI 中文摘要
我们研究首一多项式 $\Sob{n}$，它关于离散Krawtchouk内积的Sobolev型修正正交，其中在外部点 $\alpha<0$（位于二项测度支撑 $\{0,1,\dots,N\}$ 之外）处的单个点质量通过第 $j$ 阶前向差分 $\dif^{j}$ 作用（这里 $\Nzero=\{0,1,2,\dots\}$，$\Z_{+}=\{1,2,\dots\}$，$\R_{+}=(0,\infty)$）。在联合尺度 $n,N\to\infty$，$n/N\to r$ 且 $0<p<r<1$（Dominici分类中的\emph{倒数伽马}区域）下，我们证明了Mehler--Heine公式。外部质量在大度极限下精确地通过线性因子 $z-\alpha$ 存活，带有显式普适常数 $C_{\ast}$，且极限与 $\lambda>0$ 和 $j\ge1$ 均无关；同样的结论在Uvarov情形 $j=0$ 下也成立（推论~\ref{cor:uvarov}）。推导是逐点且自包含的：仅使用秩一连接公式（定理~\ref{thm:connection}）、再生核递推、Dominici逐点极限和经典三项递推；归一化的Sobolev修正项在固定 $N$ 下满足精确的仿射递推，通过两个后向窗口上的三角收缩论证并带有显式边界估计来控制，收缩比 $\varrho<1$ 恰好是条件 $r>p$。我们还讨论了定理的尺度：在这种归一化下经典项渐近可忽略，且对 $\lambda$ 的独立性涉及每个固定的 $\lambda>0$，而非在 $\lambda=0$ 处的连续性。多精度计算说明了收敛性以及 $p=0.3$，$r=3/5$，$\alpha=-2.5$，$j=2$，$\lambda=5$ 时精确值 $C_{\ast}=2.8$。
英文摘要
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 $0<p<r<1$ (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 $z-α$, with the explicit universal constant $C_{\ast}$, and the limit is independent of both $λ>0$ 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 $p=0.3$, $r=3/5$, $α=-2.5$, $j=2$, $λ=5$.
