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

TL;DR

This paper generalizes Bailey's identity for products of confluent hypergeometric functions, providing explicit formulas for various parameter cases, including new special cases, using Bailey's formula and Kummer's transformation.

Contribution

It extends Bailey's identity to broader parameter ranges and derives explicit expressions for products of hypergeometric functions, including new special cases.

Findings

Explicit formulas for hypergeometric product cases

Recovery of Bailey's original identity when parameters are zero

Introduction of new special case identities

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.