Integral representations and summations of modified Struve function

TL;DR

This paper derives new integral representations and summation formulas for the modified Struve function, connecting it with Bessel functions and hypergeometric functions, and solves related integral equations.

Contribution

It introduces novel integral formulas and summation results for the modified Struve function and related series, expanding the analytical tools available for these special functions.

Findings

New integral representations for _L_(x)

Summation formulas for Neumann, Kapteyn, and Schlf6milch series

Explicit solutions to related Fredholm integral equations

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.