Old and New Identities for Bernoulli Polynomials via Fourier Series

Luis M. Navas, Francisco J. Ruiz and Juan L. Varona

International Journal of Mathematics and Mathematical Sciences, 2012, vol. 2012, 1-14

Abstract:

The Bernoulli polynomials restricted to and extended by periodicity have n th sine and cosine Fourier coefficients of the form . In general, the Fourier coefficients of any polynomial restricted to are linear combinations of terms of the form . 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.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:129126

DOI: 10.1155/2012/129126

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