In this paper, we give an explicit formula for the first two terms of the Taylor expansion of a classical Eisenstein series of weight 2k + 1 for Γ0(q). Both the first term and the second term have interesting arithmetic interpretations. We apply the result to compute the central derivative of some Hecke L-functions. 0. Introduction. Consider the classical Eisenstein series ∑ γ∈Γ∞\ SL2(Z) Im(γτ)...

The present paper deals with the homotopy perturbation method. The question of whether the homotopy perturbation method is simply the conventional Taylor series expansion is examined. It is proven that under particular choices of the auxiliary parameters the homotopy perturbation method is indeed the Taylor series expansion of the sought solution of nonlinear equations.

It has been known since Ehrhard and Regnier’s seminal work on the Taylor expansion of λ-terms that this operation commutes with normalization: the expansion of a λ-term is always normalizable and its normal form is the expansion of the Böhm tree of the term. We generalize this result to the non-uniform setting of the algebraic λ-calculus, i.e., λ-calculus extended with linear combinations of te...

In present paper, a numerical approach for solving Cauchy type singular integral equations is discussed. Lagrange interpolation with Gauss Legendre quadrature nodes and Taylor series expansion are utilized to reduce the computation of integral equations into some algebraic equations. Finally, five examples with exact solution are given to show efficiency and applicability of the method. Also, w...

in this paper, we apply the differential transform (dt) method for finding approximate solution of the system of linear and nonlinear volterra integro-differential equations with variable coefficients, especially of higher order. we also obtain an error bound for the approximate solution. since, in this method the coefficients of taylor series expansion of solution is obtained by a recurrence r...

The Taylor expansion of λ-terms, as introduced by Ehrhard and Regnier, expresses a λ-term as a series of multi-linear terms, called simple terms, which capture bounded computations. Normal forms of Taylor expansions give a notion of infinitary normal forms, refining the notion of Böhm trees in a quantitative setting. We give the algebraic conditions over a set of normal simple terms which chara...

This article is concerned with Gaussian filtering in nonlinear continuous-discrete state-space models. We propose a novel Taylor moment expansion (TME) filter, which approximates the moments of stochastic differential equation temporal expansion. Differently from classical linearization or Itô-Taylor approaches, formed for functions directly and time variable, not by using on model. analyze the...