# On Nielsen's generalized polylogarithms and their numerical calculation

@article{Klbig1970OnNG, title={On Nielsen's generalized polylogarithms and their numerical calculation}, author={Kurt Siegfried K{\"o}lbig and Juan A. Mignaco and Ettore Remiddi}, journal={BIT Numerical Mathematics}, year={1970}, volume={10}, pages={38-73} }

The generalized polylogarithms of Nielsen are studied, in particular their functional relations. New integral expressions are obtained, and relations for function values of particular arguments are given. An Algol procedure for calculating 10 functions of lowest order is presented. The numerical values of the Chebyshev coefficients used in this procedure are tabulated. A table of the real zeros of these functions is also given.

## 110 Citations

Some New Transformation Properties of the Nielsen Generalized Polylogarithm

- MathematicsInt. J. Math. Math. Sci.
- 2014

By use of these transformation formulas presented, new fast algorithms for Nielsen generalized polylogarithm can be designed.

Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers

- Mathematics
- 2014

A survey is given on mathematical structures which emerge in multi-loop Feynman diagrams. These are multiply nested sums, and, associated to them by an inverse Mellin transform, specific iterated…

On functional equations for Nielsen polylogarithms

- Mathematics
- 2019

We derive new functional equations for Nielsen polylogarithms. We show that, when viewed modulo $\mathrm{Li}_5$ and products of lower weight functions, the weight $5$ Nielsen polylogarithm $S_{3,2}$…

The Computation of Polylogarithms

- Mathematics
- 1992

The polylogarithm function, Lip(z), is defined, and a number of algorithms are derived for its computation, valid in different ranges of its real parameter p and complex argument z. These are…

Numerical evaluation of multiple polylogarithms

- Computer Science, MathematicsComput. Phys. Commun.
- 2005

On the value of a logarithmic-trigonometric integral

- Mathematics
- 1971

A closed expression is derived for the integral ∫0π/2logncosxlogpsinxdx, wheren andp are non-negative integers. As already remarked by Nielsen in a monograph on the generalized polylogarithms…

Harmonic Sums, Polylogarithms, Special Numbers, and their Generalizations

- MathematicsArXiv
- 2013

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and…

On the all-order "-expansion of generalized hypergeometric functions with integer values of parameters

- Mathematics
- 2007

We continue our study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we apply the…

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