On Kapteyn-Kummer Series' Integral Form
Tibor K. Pog\'any, \'Arp\'ad Baricz, Anik\'o Szak\'al
TL;DR
This paper derives double integral formulas for Kapteyn-Kummer series involving confluent hypergeometric functions, unifying various special function series in a natural framework.
Contribution
It introduces new integral representations for Kapteyn-Kummer series, connecting and unifying existing results for related special function series.
Findings
Double integral expressions for Kapteyn-Kummer series derived
Unified framework for Neumann, Schlömilch, and Kapteyn-Bessel series established
Connections to recent results by Pogány, Süli, Baricz, and Jankov Maširević made
Abstract
In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's (or confluent hypergeometric ) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schl\"omilch- and Kapteyn-Bessel series recently established by Pog\'any, S\"uli, Baricz and Jankov Ma\v{s}irevi\'c.
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