Authors: Tričković, Slobodan
Stanković, Miomir 
Vidanović, Mirjana
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On the summation of Schlömilch's series
Journal: Integral Transforms and Special Functions
Volume: 31
Issue: 5
First page: 339
Last page: 367
Issue Date: 3-May-2020
Rank: M22
ISSN: 1065-2469
DOI: 10.1080/10652469.2019.1695129
Schlömilch's series is named after the German mathematician Oscar Xavier Schlömilch, who derived it in 1857 as a Fourier series type expansion in terms of the Bessel function of the first kind. However, except for Bessel functions, here we consider an expansion in terms of Struve functions or Bessel and Struve integrals as well. The method for obtaining a sum of Schlömilch's series in terms of the Bessel or Struve functions is based on the summation of trigonometric series, which can be represented in terms of the Riemann zeta and related functions of reciprocal powers and in certain cases can be brought in the closed form, meaning that the infinite series are represented by finite sums. By using Krylov's method we obtain the convergence acceleration of the trigonometric series.
Keywords: 65B10 | Bessel functions | Primary: 33C10 | Secondary: 11M06 | the Riemann zeta and Dirichlet functions
Publisher: Taylor & Francis

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