发表机构
International Chair of Mathematical Physics and Applications (ICMPA–UNESCO Chair); University of Abomey-Calavi（国际数学物理与应用 chair（ICMPA-UNESCO 讲席）; 阿波美卡拉维大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究广义Charlier多项式递推系数，通过Hankel行列式与Laguerre--Freud动力学给出其公式，并联系Painlevé V方程，数值验证大次数渐近行为。
AI 中文摘要
我们研究与离散权重 ρ_μ(k) = μ^k/(k!)^2, k∈N_0, μ>0 相关联的广义Charlier多项式的首一递推系数。递推关系 P_{n+1}(x) = (x-β_n)P_n(x) - γ_nP_{n-1}(x) 中的系数 β_n 和 γ_n 通过互补矩、行列式、离散动力学和连续可积公式来描述。我们给出了Markov函数及其Jacobi连分数的统一归一化，并利用加边行列式将对角系数 β_n 用Hankel行列式表示。从原始的Laguerre--Freud方程出发，我们推导了用于计算的局部递归系统，并将其与Toda变形联系起来。我们还回顾了与两参数广义Charlier族相关的已知Painlevé V表示，其归一化与当前特化一致。矩/Hankel构造提供了递推系数的互补矩理论实现，而局部Laguerre--Freud递归提供了独立的动力学构造。它们的比较用于数值交叉验证。数值结果与主要大次数行为 β_n ~ n, γ_n ~ μ 一致。
英文摘要
We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight \[ ρ_μ(k) = \frac{μ^k}{(k!)^2}, \qquad k\in\mathbb N_0, \qquad μ>0. \] The coefficients $β_n$ and $γ_n$ in the recurrence relation \[ P_{n+1}(x) = (x-β_n)P_n(x)-γ_nP_{n-1}(x) \] are described through complementary moment, determinant, discrete dynamical, and continuous integrable formulations. We give a consistent normalization of the Markov function and its Jacobi continued fraction, and express the recurrence coefficients in terms of Hankel determinants, using a bordered determinant for the diagonal coefficient $β_n$. Starting from the original Laguerre--Freud equations, we derive the local recursive system used for computation and relate it to the Toda deformation. We also recall the known Painlevé~V representations associated with the two-parameter generalized Charlier family, in a normalization consistent with the present specialization. Moment/Hankelconstructions provide complementary moment-theoretic realizations of the recurrence coefficients, while the local Laguerre--Freud recursion provides an independent dynamical construction. Their comparison is used for numerical cross-validation. The numerical results are consistent with the leading large-degree behavior \[ β_n\sim n, \qquad γ_n\simμ. \]
