Integral representations and summations of the modified Struve function

Abstract

It is known that Struve function $\mathbf H_\nu$ and modified Struve function $\mathbf L_\nu$ are closely connected to Bessel function of the first kind $J_\nu$ and to modified Bessel function of the first kind $I_\nu$ and possess representations through higher transcendental functions like generalized hypergeometric ${}_1F_2$ and Meijer $G$ function. Also, the NIST project and Wolfram formula collection contain a set of Kapteyn type series expansions for $\mathbf L_\nu(x)$. In this paper firstly, we obtain various another type integral representation formulae for $\mathbf L_\nu(x)$ using the technique developed by D. Jankov and the authors. Secondly, we present some summation results for different kind of Neumann, Kapteyn and Schl\"omilch series built by $I_\nu(x)$ and $\mathbf L_\nu(x)$ which are connected by a Sonin--Gubler formula, and by the associated modified Struve differential equation. Finally, solving a Fredholm type convolutional integral equation of the first kind, Bromwich--Wagner line integral expressions are derived for the Bessel function of the first kind $J_\nu$ and for an associated generalized Schl\"omilch series.