Generalization of Kummer’s second theorem with applications

Abstract

The aim of this research paper is to obtain single series expression of $$ e^{ - x/2} _1 F_1 (\alpha ;2\alpha + i;x) $$ for i = 0, ±1, ±2, ±3, ±4, ±5, where 1 F 1(·) is the function of Kummer. For i = 0, we have the well known Kummer second theorem. The results are derived with the help of generalized Gauss second summation theorem obtained earlier by Lavoie et al. In addition to this, explicit expressions of $$ _2 F_1 [ - 2n,\alpha ;2\alpha + i;2]and_2 F_1 [ - 2n - 1,\alpha ;2\alpha + i;2] $$ each for i = 0, ±1, ±2, ±3, ±4, ±5 are also given. For i = 0, we get two interesting and known results recorded in the literature. As an applications of our results, explicit expressions of $$ e^{ - x} _1 F_1 (\alpha ;2\alpha + i;x) \times _1 F_1 (\alpha ;2\alpha + j;x) $$ for i, j = 0, ±1, ±2, ±3, ±4, ±5 and $$ (1 - x)^{ - a} _2 F_1 \left( {a,b,2b + j; - \tfrac{{2x}} {{1 - x}}} \right) $$ for j = 0, ±1, ±2, ±3, ±4, ±5 are given. For i = j = 0 and j = 0, we respectively get the well known Preece identity and a well known quadratic transformation formula due to Kummer. The results derived in this paper are simple, interesting, easily established and may be useful in the applicable sciences.