It is well known that classical summation theorems such as of Gauss, Gauss second, Kummer and Bailey for the series ₂ F ₁ and Watson, Dixon and Whipple for the series ₃ F ₂ play an important role in the theory of hypergeometric and generalized hypergeometric series. Applications of these summation theorems are well known. The aim of this research paper is to provide the generalizations of the above-mentioned classical summation theorems in the most general case.

Motivated by the extension of classical Gauss′s summation theorem for the series ₂F ₁ given in the literature, the authors aim at presenting the extensions of various other classical summation theorems such as those of Kummer, Gauss′s second, and Bailey for the series ₂F ₁, Watson, Dixon and Whipple for the series ₃F ₂, and a few other hypergeometric identities for the series ₃F ₂ and ₄F ₃. As applications, certain very interesting summations due to Ramanujan have been generalized. The results derived in this paper are simple, interesting, easily established, and may be useful.

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.