Abstract We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, thus, the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis with sixth-order polynomial expansion, we demonstrate ...

In a recent paper [1] a new powerful method to calculate Feynman diagrams was proposed. It consists in setting up a Taylor series expansion in the external momenta squared. The Taylor coefficients are obtained from the original diagram by differentiation and putting the external momenta equal to zero. It was demonstrated that by a certain conformal mapping and subsequent resummation by means of...

Abstract. With z = (z1, z2, . . . , zd), we show that the Taylor coefficients of the multivariable series q(z) = zi exp(G(z)/F (z)) are integers, where F (z) and G(z) + log(zi)F (z), i = 1, 2, . . . , d, are specific solutions of certain systems of Fuchsian differential equations with maximal unipotent monodromy at z = (0, 0, . . . , 0). As consequences, we are able to prove numerous integralit...

Abstract. We show that the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)) are integers, where F(z) and G(z) + log(z)F(z) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z = 0. We also address the question of finding the largest integer u such that the Taylor coefficients of (zq(z)) are still integers. As consequences, we are...

Let a sufficiently smooth function f on [−1, 1] be sampled at n + 1 equispaced points, and let k ≥ 0 be given. An Euler–Maclaurin interpolant to the data is defined, consisting of a sum of a degree k algebraic polynomial and a degree n trigonometric polynomial, which deviates from f by O(n−k) and whose integral is equal to the order k Euler–Maclaurin approximation of the integral of f . This in...

In this paper, we study the departure process of the GI/G/1 queue. We develop a simple recursive procedure to calculate the MacLaurin series of its moments and covariances with respect to a parameter in the service time. Based on this recursive procedure the explicit formulas of the coeÆcients of these MacLaurin series can be obtained in terms of derivatives of the probability density function ...

We continue our study begun in “On the integrality of the Taylor coefficients of mirror maps” [Duke Math. J. (to appear)] of the fine integrality properties of the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)), where F(z) and G(z) + log(z)F(z) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z = 0. More precisely, we address...