In this paper, we have introduced and investigated two new subclasses of the function class Δ of bi-univalent functions defined in the open unit disk, which are associated with the generalized Hypergeometric function. Furthermore, we find estimates on the Taylor-Maclaurin coefficient | a2 | and | a3 | for the functions belonging to these new classes.

Using the Euler-MacLaurin summation formula, we give alternative proofs for the reciprocity formulas of Apostol’s Dedekind sums and generalized Hardy-Berndt sums s3,p(b, c) and s4,p(b, c). We also obtain an integral representation for each sum.

We present and comment on some details of a new analytical method for systematic improvement of the convergence of path integrals of a generic N -fold discretized theory. The new methods represents a Euler summation formula for path integrals. Keeping the first p terms in this formula improves convergence of path integrals to the continuum limit to 1/N. We have given explicit calculations up to...

In this paper we compute the generating function for the Euler characteristic of the Deligne-Mumford compactification of the moduli space of smooth n-pointed genus 2 curves. The proof relies on quite elementary methods , such as the enumeration of the graphs involved in a suitable stratification of M 2,n .

van der Corput's inequality is extended and refined by using Euler-Maclaurin formula and other analytic techniques.

In this article, van der Corput’s inequality is generalized by using the well known Euler-Maclaurin sum formula and other analytic techniques.