Integral transforms and summation of some Schlömilch series

Abstract

In this paper we present a survey of the results given in the papers [11, 12, 13, 14]. Connections between integral transforms and some Schlömilch series have also been considered. These series are represented in terms of the Riemann zeta and related functions of reciprocal powers and can be brought in so called closed form in certain cases, which means that the infinite series are represented by finite sums. As applications of our results, recursive relations of some related functions and the sums of new Schlömilch series involving the Neumann, MacDonald, Struve or Bessel functions are given as well.