The Lerch transcendent from the point of view of Fourier analysis

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

    Universidad de Salamanca

    Salamanca, España

    ROR https://ror.org/02f40zc51

  2. 2 Universidad de Zaragoza
    info

    Universidad de Zaragoza

    Zaragoza, España

    ROR https://ror.org/012a91z28

  3. 3 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

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Journal:
Journal of Mathematical Analysis and Applications

ISSN: 0022-247X

Year of publication: 2015

Volume: 431

Issue: 1

Pages: 186-201

Type: Article

DOI: 10.1016/J.JMAA.2015.05.048 SCOPUS: 2-s2.0-84937639634 WoS: WOS:000357441100013 GOOGLE SCHOLAR

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Abstract

We obtain some well-known expansions for the Lerch transcendent and the Hurwitz zeta function using elementary Fourier analytic methods. These Fourier series can be used to analytically continue the functions and prove the classical functional equations, which arise from the relations satisfied by the Fourier conjugate and flat Fourier series. In particular, the functional equation for the Riemann zeta function can be obtained in this way without contour integrals. The conjugate series for special values of the parameters yields analogous results for the Bernoulli and Apostol-Bernoulli polynomials. Finally, we give some consequences derived from the Fourier series. © 2015 Elsevier Inc.