Old and New Identities for Bernoulli Polynomials via Fourier Series

Luis M Navas,Francisco J Ruiz,Juan L Varona

doi:10.1155/2012/129126

Abstract

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

Highlights

The Bernoulli polynomials Bk x with k ∈ N ∪ {0} are defined via the generating function tetx et − 1 ∞ Bk k0 x tk k!1.1 from which one obtains the Bernoulli numbers as the values Bk Bk 0
These two properties are all that is needed to obtain the Fourier series of the 1-periodic functions which coincide with Bk x on the interval 0, 1
In more complicated situations, we believe that our method is simpler. It is the purpose of this paper to provide evidence for this claim, by obtaining, via the use of Fourier series, identities between polynomial families, some of which are new while others are well known, but are given simpler proofs

Summary

Introduction

The Bernoulli polynomials Bk x with k ∈ N ∪ {0} are defined via the generating function tetx et − 1. In more complicated situations, we believe that our method is simpler It is the purpose of this paper to provide evidence for this claim, by obtaining, via the use of Fourier series, identities between polynomial families, some of which are new while others are well known, but are given simpler proofs. It should be mentioned that new relations between classical and generalized Bernoulli polynomials and numbers are still obtainable through the use of different types of expansions see 1–4 and the references cited within These identities are similar in appearance to ours but not identical to them. The formulas appearing in this paper were checked with Maple to avoid possible mistakes in their transcription, especially the longer ones

Bernoulli and Legendre Polynomials

Bernoulli and Gegenbauer Polynomials

New Identities for Bernoulli Polynomials and Numbers

A Remark on Euler Polynomials