Some Identities Involving the Fubini Polynomials and Euler Polynomials

Guohui Chen,Li Chen

doi:10.3390/math6120300

Abstract

In this paper, we first introduce a new second-order non-linear recursive polynomials U h , i ( x ) , and then use these recursive polynomials, the properties of the power series and the combinatorial methods to prove some identities involving the Fubini polynomials, Euler polynomials and Euler numbers.

Highlights

IntroductionFor any real number x and y, the two variable Fubini polynomials Fn ( x, y) are defined by means of the following (see [1,2])
For any real number x and y, the two variable Fubini polynomials Fn ( x, y) are defined by means of the following ∞ e xt Fn ( x, y) n = ·t . ∑ 1 − y ( e t − 1) n! n =0 (1)
E0 = 1, E1 = − 12, E2 = 0, E3 = 14, E4 = 0, E5 = − 12, E6 = 0, and E2n = 0 for all positive integer n. These polynomials appear in combinatorial mathematics and play a very important role in the theory and application of mathematics, many number theory and combination experts have studied their properties, and obtained a series of interesting results

Summary

Introduction

For any real number x and y, the two variable Fubini polynomials Fn ( x, y) are defined by means of the following (see [1,2]). E0 = 1, E1 = − 12 , E2 = 0, E3 = 14 , E4 = 0, E5 = − 12 , E6 = 0, and E2n = 0 for all positive integer n These polynomials appear in combinatorial mathematics and play a very important role in the theory and application of mathematics, many number theory and combination experts have studied their properties, and obtained a series of interesting results. T. Kim et al [5] studied the properties of the Fubini polynomials Fn (y), and proved the identity n. If h = p is an odd prime, using the elementary number theory methods we deduce the following: Corollary 5. This congruence is recently obtained by Hou and Shen [12] using the different methods

Several Simple Lemmas

Proof of the Theorem