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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.

Applications of Lehmer’s Infinite Series Involving Reciprocals of the Central Binomial Coefficients