## HarmonicSums

### A Mathematica Package for dealing with Harmonic Sums, Generalized Harmonic Sums and Cyclotomic Sums and their related Integral Representations

### Authors

`HarmonicSums`package by Jakob Ablinger allows to deal with nested sums such as harmonic sums, S-sums, cyclotomic sums and cyclotmic S-sums as well as iterated integrals such as harmonic polylogarithms, multiple polylogarithms and cyclotomic polylogarithms in an algorithmic fashion. The package can calculte the Mellin transformation of the iterated integrals in terms of the nested sums and it can compute integral representations of the nested sums. The package can be used to compute algebraic and structural relations between the nested sums as well as between the the iterated integrals and connected to it the package can find relations between the nested sums at infinity and the iterated integrals at one. In addition the package provides algorithms to represent expressions involving the nested sums and iterated integrals in terms of basis representations. Moreover, the package allows to compute (asymptotic) expansions of the nested sums and iterated integrals and it contains an algorithm which rewrites certain types of nested sums into expressions in terms of cyclotomic S-sums.

### Registration and Legal Notices

The source code for this package is password protected. To get the password send an email to Peter Paule. It will be given for free to all researchers and non-commercial users. Copyright © 2007–2019 The RISC Combinatorics Group, Austria — all rights reserved. Commercial use of the software is prohibited without prior written permission.### A Note on Encoded Files

This package contains one or more Mathematica input files which are encoded. Those files cannot be read or modified directly as plain text, but can be loaded into Mathematica just like any normal input file (i.e., with**<<"**or

*file*"**Get["**). There is no need (and also no way) to decode them by using additional software or a special key.

*file*"]### The Package

The package is contained in the Mathematica input file- HarmonicSums.m (encoded)

`HarmonicSumsFunctionList[]`to get a description of the available commands. Thanks to the tremendous effort of Abilio De Freitas all the available commands of

`HarmonicSums`now have usage messages. These messages are summarized in the notebook UsageMessages.nb. The following precomputed tables are available in order to extend the functionality of the package and are used by the functions

`ReduceToBasis`and

`ReduceConstants`:

- RelTabSinfH1.m
- RelTabCycloH6tow4.m
- RelTabCycloH4ToSinf.m
- RelTabSinfH1upto8.m
- RelTabCSinf.m
- RelTab.m
- RelTabStat.m
- RelTabDiff.m
- RelTabHalf.m
- RelTabDiffHalf.m

### Discovering and Proving Infinite Pochhammer Sum Identities

The notebook PochhammerSums.nb accompanies the article- J. Ablinger. Discovering and Proving Infinite Pochhammer Sum Identities. arXiv:1902.11001 [math.CO].

### Proving two conjectural series for ζ(7) and discovering more series for ζ(7).

The notebook ProofOfZeta7Identities.nb and the relation table HLogCyclo3tow7.m accompany the article- J. Ablinger. Proving two conjectural series for ζ(7) and discovering more series for ζ(7). arXiv:1908.06631 [math.CO].

### Literature

The theoretical background of the algorithms implemented in HarmonicSums is described in- Computer Algebra Algorithms for Special Functions in Particle Physics (PhD Thesis). RISC, Johannes Kepler University, April 2012. [pdf]
- A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics (Diploma Thesis). RISC, Johannes Kepler University, February 2009. [pdf]

- J. Ablinger. A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics. Johannes Kepler University. Diploma Thesis. February 2009. arXiv:1011.1176 [math-ph].
- J. Ablinger. Computer Algebra Algorithms for Special Functions in Particle Physics. Johannes Kepler University. PhD Thesis. April 2012.
- J. Ablinger. Computing the Inverse Mellin Transform of Holonomic Sequences using Kovacic's Algorithm. PoS RADCOR2017, 001, 2017. arXiv:1801.01039 [cs.SC].
- J. Ablinger. Inverse Mellin Transform of Holonomic Sequences. PoS LL 2016, 067, 2016. arXiv:1606.02845 [cs.SC].
- J. Ablinger. The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums. Loops and Legs in Quantum Field Theory - LL 2014. arXiv:1407.6180 [cs.SC].
- J. Ablinger, J. Blümlein and C. Schneider. Generalized Harmonic, Cyclotomic, and Binomial Sums, their Polylogarithms and Special Numbers. ACAT 2013 , J. Phys.: Conf. Ser 523/012060. arXiv:1310.5645 [math-ph]
- J. Ablinger. Discovering and Proving Infinite Binomial Sums Identities. Experimental Mathematics 26, 2017. arXiv:1507.01703 [math.CO].
- J. Ablinger. An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences. Loops and Legs in Quantum Field Theory - LL 2018. PoS(LL2018)
- J. Ablinger. Discovering and Proving Infinite Pochhammer Sum Identities. Experimental Mathematics, 2019. arXiv:1902.11001 [math.CO].
- J. Ablinger, J. Blümlein and C. Schneider. Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms. J. Math. Phys. 54 (2013) arXiv:1302.0378 [math-ph].
- J. Ablinger, J. Blümlein, C.G. Raab, C. Schneider. Iterated Binomial Sums and their Associated Iterated Integrals. J. Math. Phys. 55 (2014) arXiv:1407.1822 [hep-th].
- J. Ablinger, J. Blümlein and C. Schneider. Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials. J. Math. Phys. 52 (2011) 102301. arXiv:1105.6063 [math-ph].
- J. Blümlein and S. Kurth. Harmonic sums and Mellin transforms up to two loop order. Phys. Rev. D 60 (1999) 014018. hep-ph/9810241 [hep-ph].
- J. Blümlein. Algebraic relations between harmonic sums and associated quantities. Comput. Phys. Commun. 159 (2004). arXiv:hep-ph/0311046 [hep-ph].
- J. Blümlein. Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5. Comput. Phys. Commun. 180 (2009) 2218. arXiv:0901.3106 [hep-ph].
- E. Remiddi and J. A. M. Vermaseren. Harmonic polylogarithms. Int. J. Mod. Phys. A 15 (2000) 725. [hep-ph/9905237].
- J. A. M. Vermaseren. Harmonic sums, Mellin transforms and integrals. Int. J. Mod. Phys. A 14 (1999) 2037. [hep-ph/9806280].