# A fast and simple algorithm for the computation of the Lerch transcendent

## TL;DR

This paper presents a fast, reliable algorithm using Gauss-Laguerre quadrature for computing the Lerch transcendent, including error estimates and a MATLAB implementation, enabling arbitrary precision calculations.

## Contribution

It introduces a novel, efficient algorithm for Lerch transcendent computation with error control and practical MATLAB code, improving upon existing methods.

## Key findings

- Error estimates enable precise quadrature node selection
- Algorithm achieves high accuracy with fewer nodes
- Numerical tests confirm reliability and efficiency

## Abstract

This paper deals with the computation of the Lerch transcendent by means of the Gauss-Laguerre formula. An a priori estimate of the quadrature error, that allows to compute the number of quadrature nodes necessary to achieve an arbitrary precision, is derived. Exploiting the properties of the Gauss-Laguerre rule and the error estimate, a truncated approach is also considered. The algorithm used and its Matlab implementation are reported. The numerical examples confirm the reliability of this approach.