Old and new identities for Bernoulli polynomials via Fourier series

  1. Navas, L.M. 1
  2. Ruiz, F.J. 2
  3. Varona, J.L. 3
  1. 1 Universidad de Salamanca

    Universidad de Salamanca

    Salamanca, España

    ROR https://ror.org/02f40zc51

  2. 2 Universidad de Zaragoza

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

  3. 3 Universidad de La Rioja

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

International Journal of Mathematics and Mathematical Sciences

ISSN: 0161-1712

Year of publication: 2012

Volume: 2012

Pages: 1-15

Type: Article

DOI: 10.1155/2012/129126 SCOPUS: 2-s2.0-84864925720 GOOGLE SCHOLAR lock_openOpen access editor

More publications in: International Journal of Mathematics and Mathematical Sciences


The Bernoulli polynomials B k restricted to [0, 1) and extended by periodicity have nth sine and cosine Fourier coefficients of the form C k/n k. In general, the Fourier coefficients of any polynomial restricted to [0, 1) are linear combinations of terms of the form 1/n k. 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. Copyright © 2012 Luis M. Navas et al.