# Summation of Series and Gaussian Quadratures

## Abstract

In 1985, Gautschi and the author constructed Gaussian quadrature formulae on (0, +∞) involving Einstein and Fermi functions as weights and applied them to the summation of slowly convergent series which can be represented in terms of the derivative of a Laplace transform, or in terms of the Laplace transform itself. A problem that may arise in this procedure is the determination of the respective inverse Laplace transform. For the class of slowly convergent series, \(\sum\limits_{k = 1}^{ + \,\infty } {{{\left( { \pm 1} \right)}^k}} {a_k}with\,{a_k} = {k^{v - 1}}R\left( k \right)
\)where 0 < v ≤ 1 and *R*(s) is a rational function, Gautschi recently solved this problem. In the present paper, using complex integration and constructing Gauss-Christoffel quadratures on (0, +∞) with respect to the weight functions *w* _{1} (*t*) = 1/cosh^{2} *t* and *w* ^{2} (*t*) = sinh *t*/ cosh^{2} *t*, we reduce the series \(\sum\limits_{k\, = \,m}^{ + \,\infty } {f\left( k \right)} \,and\,{\sum\limits_{k\, = \,m}^{ + \,\,\infty } {\left( { - \,1} \right)} ^k}\,f\,\left( k \right)\) to weighted integrals of *f* involving weights *w* _{1} and *w* _{2}, respectively. We illustrate this method with a few numerical examples.

## Keywords

Relative Error Weight Function Orthogonal Polynomial Gaussian Approximation Gaussian Quadrature## Preview

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