Integral representations and summations of modified Struve function

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 ${}_1{F}_2$ 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.