Generalization of two theorems due to Ramanujan

Abstract

The aim in this paper is to provide generalizations of two interesting entries in Ramanujan's notebooks that relate sums involving the derivatives of a function φ(t) evaluated at 0 and 1. The generalizations obtained are derived with the help of expressions for the Gauss hypergeometric function ₂ F ₁(−n, a; 2a+j; 2) for non-negative integer n and j=0,±1, …,±5 given very recently by Kim et al. [Generalizations of Kummer's second theorem with applications, Comput. Math. Math. Phys. 50(3) (2010), pp. 387–402] and extension of Gauss’ summation theorem available in the literature. Several special cases that are closely related to Ramanujan's results are also given.

Keywords:

• hypergeometric series
• generalized Gauss summation theorem
• Ramanujan's sum
• sums of Hermite polynomials

Mathematics Subject Classification:

• 33C15
• 33C20

Acknowledgements

Y.S. Kim acknowledges the support of the Wonkwang University Research Fund (2012). A.K. Rathie thanks the Department of Science and Technology, Government of India, for providing financial assistance and the Centre for Mathematical Sciences for providing all necessary facilities.

Notes

When a=0,−1,−2, … a limiting procedure has to be applied to the ₂ F ₂ function.