AI 中文总结
该研究分析了受秩一端点质量扰动的Charlier和Meixner多项式的有限差分的Mehler--Heine渐近行为，证明其极限轮廓与经典多项式一致，揭示了控制其渐近稳定性的尺度分离机制。
AI 中文摘要
点质量扰动和有限差分操作会改变离散正交多项式的有限次结构，但它们在Mehler--Heine尺度下的联合效应并不直观。我们研究了首一Charlier和Meixner族在支撑端点处受固定质量Aδ₀的纯Uvarov修正的情况，并确定了它们任意固定阶的前向和后向差分的渐近行为。利用秩一连接公式，我们证明了在Charlier情形下，扰动系数按阶乘衰减，而在Meixner情形下则按指数衰减并带有代数前因子。在两个族中，这种衰减都快于n⁻¹的任意代数幂，因此抑制了任意固定数量有限差分所产生的增长。因此，对于每个固定的A≥0和k∈ℕ₀，质量修正族和经典族在复平面ℂ上具有相同的局部均匀Mehler--Heine极限。前向差分保留了倒数伽马轮廓，仅相差因子(-1)ᵏ；而后向差分将极限自变量平移k，使极限零点格从ℕ₀移至k+ℕ₀。我们还推导了公共前向极限的一阶平移方程，并刻画了其整个解空间。复平面图像和实轴计算说明了预测的极限轮廓、两个零点格以及固定端点质量的渐近消失。这些结果确定了一种尺度分离机制，用于控制Charlier和Meixner族在秩一端点扰动下的固定阶渐近稳定性。
英文摘要
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 $Aδ_{0}$ 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.
