Integral and computational representation of summation which extends a Ramanujan's sum

Tibor K. Pogány, Arjun K. Rathie, Shoukat Ali

A generalized sum, which contains Ramanujan's summation formula recorded in Hardy's article [G.H. Hardy, A chapter from Ramanujan's notebook, Proc. Camb. Phil. Soc. 21 (1923), 492-503] as a special case, has been represented in the form of Mellin-Barnes type contour integral. A computational representation formula is derived for this summation in terms of the unified Hurwitz-Lerch Zeta function.