On generalization of Bailey's identity involving product of generalized hypergeometric series

Abstract

The aim of this research paper is to obtain explicit expressions of (i) ${}_1F_1 \left[\begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right]. {}_1F_1\left[ \begin{array}{c} \beta \\ 2\beta + j \end{array} ; x \right]$ (ii) ${}_1F_1 \left[ \begin{array}{c} \alpha \\ 2\alpha - i \end{array} ; x \right] . {}_1F_1 \left[ \begin{array}{c} \beta \\ 2\beta - j \end{array} ; x \right]$ (iii) ${}_1F_1 \left[ \begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right] . {}_1F_1 \left[\begin{array}{c} \beta \\ 2\beta - j \end{array} ; x \right]$ in the most general form for any $i,j=0,1,2,\ldots$ For $i=j=0$, we recover well known and useful identity due to Bailey. The results are derived with the help of a well known Bailey's formula involving products of generalized hypergeometric series and generalization of Kummer's second transformation formulas available in the literature. A few interesting new as well as known special cases have also been given.