On extensions of two results due to Ramanujan

Yong Sup Kim; Arjun Kumar Rathie; Richard Bruce Paris

doi:10.3906/mat-1909-59

On extensions of two results due to Ramanujan

Yong Sup Kim^*, Arjun Kumar Rathie, Richard Bruce Paris

^*Corresponding author for this work

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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 φ(t) evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating ₃F₂ hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating ₃F₂(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.

Original language	English
Pages (from-to)	348-355
Number of pages	8
Journal	Turkish Journal of Mathematics
Volume	44
Issue number	2
Early online date	28 Dec 2019
DOIs	https://doi.org/10.3906/mat-1909-59
Publication status	Published - 17 Mar 2020

Keywords

Hypergeometric series
Ramanujan's sum
Sums of Hermite polynomials

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Cite this

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AB - 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 φ(t) evaluated at 0 and 1. The generalization obtained is derived with the help of expressions for the sum of terminating 3F2 hypergeometric functions of argument equal to 2, recently obtained by Kim et. al. [Two results for the terminating 3F2(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.

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On extensions of two results due to Ramanujan

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