Abstract
The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function F q + 1 q z for q = 2 , 3 , 4 , 5 . This is achieved by means of separating the generalized hypergeometric function F q + 1 q z ( q = 2 , 3 , 4 , 5 ) into even and odd components together with the use of several known infinite series involving reciprocals of the central binomial coefficients obtained earlier by Lehmer.
Highlights
Lehmer’s Series Involving Central Binomial CoefficientsN ≥ k, n < k, ð1Þ for nonnegative integers n and k
The main objective of this paper is to establish several new closed-form evaluations of the generalized hypergeometric function
The central binomial coefficients are defined by ð2nÞ! ðn!Þ2 ðn. It is well-known that the binomial and reciprocal of binomial coefficients play an important role in many areas of mathematics
Summary
N ≥ k, n < k, ð1Þ for nonnegative integers n and k. The central binomial coefficients are defined by. It is well-known that the binomial and reciprocal of binomial coefficients play an important role in many areas of mathematics (including number theory, probability, and statistics). The sums containing the central binomial coefficients and reciprocals of the central binomial coefficients have been studied for a long time. Many facts about the central binomial coefficients and the reciprocals of the central binomial coefficients can be found in the book of Koshy [12]. Gould [13] has collected numerous identities involving central binomial coefficients. 2n ð3Þ n where an are of very simple functions of n and deduced several interesting series involving the central binomial coefficients and reciprocalspofffiffi the central binomial coefficients, with Golden ratio φ = ð 5 + 1Þ/2 as follows:. We conclude this section by remarking that the results (4), (5), (6), (7), (8), (9), (10), (11), (12), ð16Þ (13), (14), (15), (16), (17), (18), (19), (20), (21), (22), (23), (24), (25), (26), AND (27) will be written in terms of a generalized hypergeometric function
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