贝努利数和富比尼多项式及其算法
Bernoulli Numbers and Fubini Polynomials and an Algorithm for Them
DOI: 10.12677/pm.2024.1411394, PDF,    国家自然科学基金支持
作者: 刘伟明：北京石油化工学院数理系，北京；于 快, 程晓亮：吉林师范大学数学与计算机学院，吉林 四平
关键词: 贝努利数；富比尼多项式；算法；恒等式；Bernoulli Numbers； Fubini Polynomials； Algorithm； Identities
摘要: 文章提出一种新的算法，选用适当的初始值，则相应地得到贝努利数、多贝努利数以及富比尼多项式等著名数和多项式。研究了该算法的原理，给出并证明了算法生成的无穷阶矩阵的第0列的显式计算公式，以及第0列到第0行的逆变换公式以及生成函数的表示，它们可用于建立恒等式及封闭计算公式。作为应用，得到有关这些著名数和多项式的一些封闭的计算公式以及恒等式。
Abstract: A new algorithm for Bernoulli numbers, Poly-Bernoulli numbers and Fubini polynomials is given by using appropriate initial values in this paper. The principle of the algorithm is discussed, and the explicit formula for the 0-th column of the matrix generated by the algorithm, the inverse transform and their expression by generating funtions from the 0-th column to the 0-th row are given, which can be used to establish identities and closed formulae. As applications some identities and closed formulae for these famous numbers and polynomials are given.
文章引用：刘伟明, 于快, 程晓亮. 贝努利数和富比尼多项式及其算法[J]. 理论数学, 2024, 14(11): 268-279. https://doi.org/10.12677/pm.2024.1411394

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