Abstract
Very recently, Masjed-Jamei & Koepf [Some summation theorems for generalized hypergeometric functions, Axioms, 2018, 7, 38, 10.3390/axioms 7020038] established some summation theorems for the generalized hypergeometric functions. The aim of this paper is to establish extensions of some of their summation theorems in the most general form. As an application, several Eulerian-type and Laplace-type integrals have also been given. Results earlier obtained by Jun et al. and Koepf et al. follow special cases of our main findings.
Highlights
The well-known and useful Pochhammer symbol (or the shifted or the raised factorial, since (1)n = n!) denoted by (a)n for any complex number a is defined by (a)n =a(a + 1) . . . (a + n − 1) ; (n ∈ N and a ∈ C); (n = 0 and a ∈ C \ {0}) (1.1) 65 Page 2 of 19
For interesting results by employing the above mentioned classical summation theorems, we refer to a paper by Bailey [3]
For generalizations of the above mentioned classical summation theorems (1.5), (1.6) and (1.7), we refer to research papers by Lavoie, et al [9,10,11] and Rakha and Rathie [16]
Summary
We observe that the first 3F2 appearing on the right-hand side can be evaluated with the help of the extended Bailey’s summation theorem (1.11), and we arrive at the right-hand side of (5.1). Vol 76 (2021) A Study of Extensions of Classical Summation Theorems. In (5.2) and (5.3), if we take d = b, we recover known results due to Masjed-Jamei and Koepf [12].
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