Recent results relevant to the evaluation of infinite series

Prespacetime Journal, 2013

In this paper, Abel summation method is applied to evaluate infinite series and divergent integrals. Several examples of how one can obtain regularizations are given.

The European Physical Journal Plus, 2016

A method is suggested for treating the well-known deficiency in the use of Padé approximants that are well suited for approximating rational functions, but confront problems in approximating irrational functions. We develop the approach of self-similarly corrected Padé approximants, making it possible to essentially increase the class of functions treated by these approximants. The method works well even in those cases, where the standard Padé approximants are not applicable, resulting in divergent sequences. Numerical convergence of our method is demonstrated by several physical examples.

Journal of Approximation Theory, 1988

2007

Asymptotic approximations (n → ∞) to the truncation errors rn = − ∞ ν=0 aν of infinite series ∞ ν=0 aν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation ∆rn = an+1. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered-the Gaussian hypergeometric series 2F1(a, b; c; z) and the divergent asymptotic inverse power series for the exponential integral E1(z)-the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

Mathematics of Computation, 1994

We deal with a method of enhanced convergence for the approximation of analytic functions. This method introduces conformai transformations in the approximation problems, in order to help extract the values of a given analytic function from its Taylor expansion around a point. An instance of this method, based on the Euler transform, has long been known; recently we introduced more general versions of it in connection with certain problems in wave scattering. In §2 we present a general discussion of this approach.

Rocky Mountain Journal of Mathematics, 1974

We define the Padé approximant, invented by Jacobi [ 12], to the formal power series /(x), by the equations rr,w, _ W \1) (2) (3) L1^1J "• QM(X) Q M (x)f(x)-P L {x) = 0(x « QM(0) = 1 where P L and Q M are polynomials of degree L and M respectively. This definition differs from the classical one of Frobenius [9] and Padé [14], in the use of (3). Under our definition the Padé approximants do not always exist, but an infinite number on each row, column, and diagonal of the Padé table always do exist [4]. The diagonal, L= M, Padé approximants satisfy the following invariance theorem, THEOREM (INVARIANCE). If PM(X)IQM(X) is the [M/M] Padé approximant to f(x), and C + Df (0) ^ 0, then

1997

Physics Reports, 2007

This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

Journal of Computational and Applied Mathematics, 1980

Recently McCabe and Murphy have considered the two-point Pad6 approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.