An Extension of the Triple Hypergeometric Series by Exton

Abstract

Abstract. The aim of this paper is to extend a number of transformation formulas for the four X 4 , X 5 , X 7 , and X 8 among twenty triple hypergeometric series X1 to X20 introduced earlier by Exton. The results are derived from the generalized Kummer's theorem and Dixon's theorem obtained earlier by Lavoie et al..

On the Connections between Certain Class of New Quadruple and Known Triple Hypergeometric Series

“…In 1982 Exton [8] published a very interesting and useful research paper in which he encountered a number of triple hypergeometric functions of second order whose series representations involve such products as  in their kernels. It is not out of place to mention here that Exton's functions 1 X to 20 X have been studied a lot until today; see, for example, the works [10,[3][4][5][6][11][12][13]. Moreover, Exton [8] presented a large number of very interesting transformation formulas and reducible cases with the help of two known results which are called in the literature Kummer's first and second transformations or theorems.…”

Generalization of a Transformation Formula for the Exton's Triple Hypergeometric Series X12and X17

“…It is not out of place to mention here that the Exton's functions X\ to X20 have been studied a lot until today, see, for example, the works [2], [3], [4], [5], [8], [9], [10] and [11]. Moreover, Exton [6] presented a large number of very interesting transformation formulas and reducible cases with the help of two known results which are called in the literature as Hummer's first and second transformations or theorems.…”

CERTAIN INTEGRAL REPRESENTATIONS OF EULER TYPE FOR THE EXTON FUNCTION X8

“…, 20) to investigate their twenty Laplace integral representations which include the confluent hypergeometric functions 0 F 1 , 1 F 1 , a Humbert function Ψ 1 , a Humbert function Φ 2 in their kernels. The Exton functions X i have been studied a lot until today, for example, see [2,5,6,7,8,9,10]. Here, we choose to investigate the Exton function X 8 to present (presumably new) 14 integral representations of Euler type whose kernels contain the Exton function X 2 itself, the Horn's function H 4 , Gauss hypergeometric function F = 2 F 1 , and Lauricella hypergeometric function F C .…”