Transformation formulas

Abstract

In this chapter we study some transformations that give interesting relations between the Ramanujan summation and other summations. In the first section we examine the Borel summability of the series deduced from the Euler-MacLaurin formula. In the second section we use finite differences and Newton series to give a convergent version of the Ramanujan summation which generalizes the classical Laplace-Gregory formula. In the third section we use the Euler-Boole summation formula to link the Ramanujan summation of even and odd terms of a series with the Euler summation of the corresponding alternate series.