关于高阶富比尼多项式的一些新的恒等式

Some New Identities on Fubini Polynomials of Higher Order

摘要: 探讨高阶富比尼多项式的性质。利用生成函数、建立偏微分方程以及复分析等方法和技巧，给出关于高阶富比尼多项式的一些新的递推关系、封闭计算公式及恒等式。特别是包括：高阶富比尼多项式与贝努利数的关系式，它满足的微分–差分方程以及高阶富比尼多项式的一种无穷项和的计算公式。作为新的恒等式的应用，研究了高阶富比尼多项式的同余性，特别是将学者Diagana等深入使用p-adic变换和p-adic积分理论而得到的关于富比尼数的恒等式及同余性结果，推广到更一般的富比尼多项式情形，并提出一个微分算子的计算问题。

Abstract: The properties of Fubini polynomials of higher order are investigated. Using generating functions, partial-differential equations, complex analysis and related techniques, we establish several new recurrence relations, closed-form expressions and identities for these polynomials. In particular, we derive explicit connections between Fubini polynomials and Bernoulli numbers of higher order, a differential-difference equation satisfied by Fubini polynomials of higher order, and an infinite-series evaluation formula. As applications of the new identities we study congruences for Fubini polynomials of higher order, extending the identities and congruences for Fubini numbers obtained by Diagana et al. via p-adic Laplace transforms and p-adic integration to the more general setting of Fubini polynomials. Finally, we propose a computational problem involving a specific differential operator.