Applications of Extended Kummer’s Summation Theorem

Xiaoxia Wang, Arjun K. Rathie (), Eunyoung Lim and Hwajoon Kim ()
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Xiaoxia Wang: Department of Mathematics, Shanghai University, Shanghai 200444, China
Arjun K. Rathie: Department of Mathematics, Vedant College of Engineering & Technology, Rajasthan Technical University, Bundi 323021, Rajasthan, India
Eunyoung Lim: Department of IT Engineering, Kyungdong University, Yangju 11458, Republic of Korea
Hwajoon Kim: Department of IT Engineering, Kyungdong University, Yangju 11458, Republic of Korea

Mathematics, 2024, vol. 12, issue 19, 1-14

Abstract: In the theory of hypergeometric series and generalized hypergeometric series, classical summation theorems, such as the two of Gauss and those of Kummer and Bailey for the series F 1 2 ; those of Watson, Dixon, Whipple, and Saalschutz for the series F 2 3 ; and others, play a key role. Applications of these classical summation theorems are well known. Berndt pointed out that a large number of interesting summations (including Ramanujan’s summations and the Gregory–Leibniz pi summation) can be obtained very quickly by employing the above-mentioned classical summation theorems. Also, several interesting results involving products of generalized hypergeometric series have been obtained by Bailey by employing the above-mentioned classical summation theorems. Recently, the above-mentioned classical summation theorems have been generalized and extended. In our present investigations, our aim is to demonstrate the applications of the extended Kummer’s summation theorem in establishing (i) extensions of Gauss’s second summation theorem and Bailey’s summation theorem; (ii) extensions of several summations (including Ramanujan’s summations); (iii) extensions of several results involving products of generalized hypergeometric series; and (iv) an extension of classical Dixon’s summation theorem. As special cases, we recover several known summations (including several Ramanujan summations and the Gregory–Leibniz pi summation) and various results involving products of generalized hypergeometric series due to Bailey.

Keywords: Gauss summation theorem; Bailey summation theorem; Kummer summation theorem; extension of Kummer summation theorem; Dixon summation theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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