关于离散查理尔和迈克斯纳索伯列夫型正交多项式的生成函数与梅勒 - 海涅公式
On generating functions and Mehler--Heine formulas for discrete Charlier and Meixner Sobolev-type orthogonal polynomials
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中文总结 AI 辅助
研究离散索伯列夫型查理尔和迈克斯纳正交多项式，开发首个统一生成函数框架，推导相关生成函数及迭代向前差分的，通过渐近分析揭示外部索伯列夫扰动特性，扩展了该领域分析理论并建立多方面联系。
中文摘要 AI 辅助
生成函数是正交多项式理论中重要的分析工具。离散索伯列夫正交多项式虽有广泛发展，但与任意阶向前差分\(j\geq 1\)和外部质量点\(\alpha<0\)相关的索伯列夫型查理尔和迈克斯纳族尚无通用生成函数理论。本文为此开发了首个统一框架。从显式连接公式出发，推导了索伯列夫型多项式及其迭代向前差分的生成函数，用于建立新的梅勒 - 海涅公式。渐近分析表明，外部索伯列夫扰动产生一个收敛到质量点的例外零点，其余零点保持经典渐近分布，且极限梅勒 - 海涅函数与索伯列夫质量参数和向前差分算子阶数无关，揭示了高阶离散索伯列夫扰动的普遍性现象。这些结果扩展了离散索伯列夫正交多项式的分析理论，建立了生成函数、梅勒 - 海涅渐近性与离散索伯列夫型查理尔和迈克斯纳族零点渐近分布之间的直接联系。
英文摘要
Generating functions are among the most important analytical tools in the theory of orthogonal polynomials, providing a unified framework for deriving structural identities, asymptotic expansions, and zero distributions. However, despite the extensive development of discrete Sobolev orthogonal polynomials, no general generating-function theory has been available for the Sobolev-type Charlier and Meixner families associated with arbitrary-order forward differences $j\geq 1$ and an exterior mass point $α<0$. In this paper, we develop the first unified generating-function framework for these families. Starting from explicit connection formulas, we derive generating functions for the Sobolev-type polynomials and their iterated forward differences, which serve as the main analytical tool for establishing new Mehler--Heine formulas. The resulting asymptotic analysis shows that an exterior Sobolev perturbation generates exactly one exceptional zero converging to the mass point, while the remaining zeros preserve the classical asymptotic distribution. Moreover, the limiting Mehler--Heine functions are independent of both the Sobolev mass parameter and the order of the forward difference operator, revealing a universality phenomenon for higher-order discrete Sobolev perturbations. These results considerably extend the analytical theory of discrete Sobolev orthogonal polynomials and establish a direct connection between generating functions, Mehler--Heine asymptotics, and the asymptotic distribution of the zeros for the discrete Sobolev-type Charlier and Meixner families.
