On a representation of Humbert's double hypergeometric series $\Phi_3$ in a series of Gauss's $_2F_1$ function

TL;DR

This paper presents an alternative proof for a recent series representation of Humbert's double hypergeometric series $\

Contribution

It offers a new proof of a recent series expansion of Humbert's $\

Findings

Provides an alternative proof of the series representation.

Includes interesting special cases of the series.

Enhances understanding of hypergeometric series relations.

Abstract

Very recently a new series representation of Humbert's double hypergeometric series $\Phi_3$ in series of Gauss's $_2F_1$ function was given by one of us. The aim of this short research note is to provide an alternative proof of the result. A few interesting special cases are also given.