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Integral representations and summations of modified Struve function

It is known that Struve function H_nu and modified Struve function 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 L_nu(x). In this paper firstly, we obtain various another type integral representation formulae for 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 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.

Modified Struve function ; Bessel function and modified Bessel function of the first kind ; Neumann ; Kapteyn and Schl\

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