The Lerch transcendent from the point of view of Fourier analysis
- Navas, L.M. 1
- Ruiz, F.J. 2
- Varona, J.L. 3
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1
Universidad de Salamanca
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2
Universidad de Zaragoza
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3
Universidad de La Rioja
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ISSN: 0022-247X
Año de publicación: 2015
Volumen: 431
Número: 1
Páginas: 186-201
Tipo: Artículo
Otras publicaciones en: Journal of Mathematical Analysis and Applications
Resumen
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.