On extensions of two results due to Ramanujan

Authors : Yong Sup Kim, Arjunkumar Rathie, Rıchard B. Paris

Pages : 348-355

Doi::10.3906/mat-1909-59

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Publication Date : 9999-12-31

Article Type : Makaleler

Abstract :The aim in this note is to provide a generalization of an interesting entry in Ramanujan's notebooks that relate sums involving the derivatives of a function $\varphi\left t\right $ evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating ${}_3F_2$ hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating ${}_3F_2$ 2 with applications, Bulletin of the Korean Mathematical Society 2012; 49: 621-633]. Several special cases are given. In addition we generalize a summation formula to include integral parameter differences.
Keywords : Hypergeometric series, Ramanujan's sum, sums of Hermite polynomials