Computer Algebra for Combinatorics

Computer algebra for enumerative combinatorics and related fields like symbolic integration and summation, number theory (partitions, q-series, etc.), and special functions, incl. particle physics.

Computer Algebra for Combinatorics at RISC is devoted to research that combines computer algebra with enumerative combinatorics and related fields like symbolic integration and summation, number theory (partitions, q-series, etc.), and special functions, including particle physics. For further details see the research groups

Software

Bibasic Telescope

A Mathematica Implementation of a Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation

This package is part of the RISCErgoSum bundle. pqTelescope is a Mathematica implementation of a generalization of Gosper’s algorithm to indefinite bibasic hypergeometric summation. The package has been developed by Axel Riese, a former member of the RISC Combinatorics group. ...

Authors: Axel Riese
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DiffTools

A Mathematica Implementation of several Algorithms for Solving Linear Difference Equations with Polynomial Coefficients

DiffTools is a Mathematica implementation for solving linear difference equations with polynomial coefficients. It contains an algorithm for finding polynomial solutions (by Marko Petkovsek), the algorithm by Sergei Abramov for finding rational solutions, the algorithm of Mark van Hoeij for ...

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Engel

A Mathematica Implementation of q-Engel Expansion

This package is part of the RISCErgoSum bundle. Engel is a Mathematica implementation of the q -Engel Expansion algorithm which expands q-series into inverse polynomial series. Examples of q-Engel Expansions include the Rogers-Ramanujan identities together with their elegant generalization by ...

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GenOmega

A Mathematica Implementation of Guo-Niu Han's General Algorithm for MacMahon's Partition Analysis

This package is part of the RISCErgoSum bundle. GenOmega is a Mathematica implementation of Guo-Niu Han’s general Algorithm for MacMahon’s Partition Analysis carried out by Manuela Wiesinger, a master student of the RISC Combinatorics group. Partition Analysis is a computational ...

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Guess

A Mathematica Package for Guessing Multivariate Recurrence Equations

This package is part of the RISCErgoSum bundle. The Guess package provides commands for guessing multivariate recurrence equations, as well as for efficiently guessing minimal order univariate recurrence, differential, or algebraic equations given the initial terms of a sequence or ...

Authors: Manuel Kauers
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HarmonicSums

The 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. ...

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HolonomicFunctions

A Mathematica Package for dealing with Multivariate Holonomic Functions, including Closure Properties, Summation, and Integration

This package is part of the RISCErgoSum bundle. The HolonomicFunctions package allows to deal with multivariate holonomic functions and sequences in an algorithmic fashion. For this purpose the package can compute annihilating ideals and execute closure properties (addition, multiplication, substitutions) ...

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ModularGroup

ModularGroup.m is a Mathematica package which has been developed in the course of the diploma thesis Computer Algebra and Analysis: Complex Variables Visualized, carried out at the Research Institute for Symbolic Computation (RISC) of the Johannes Kepler University Linz ...

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MultiSum

A Mathematica Package for Proving Hypergeometric Multi-Sum Identities

This package is part of the RISCErgoSum bundle. MultiSum is a Mathematica package for proving hypergeometric multi-sum identities. It uses an efficient generalization of Sister Celine’s technique to find a homogeneous polynomial recurrence relation for the sum. The package has ...

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Omega

A Mathematica Implementation of Partition Analysis

Omega is a Mathematica implementation of MacMahon’s Partition Analysis carried out by Axel Riese, a Postdoc of the RISC Combinatorics group. It has been developed together with George E. Andrews and Peter Paule within the frame of a project initiated ...

Authors: Axel Riese
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OreSys

A Mathematica Implementation of several Algorithms for Uncoupling Systems of Linear Ore Operator Equations

This package is part of the RISCErgoSum bundle. OreSys is a Mathematica package for uncoupling systems of linear Ore operator equations. It offers four algorithms for reducing systems of differential or (q-)difference equations to higher order equations in a single ...

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qMultiSum

A Mathematica Package for Proving q-Hypergeometric Multi-Sum Identities

This package is part of the RISCErgoSum bundle. qMultiSum is a Mathematica package for proving q-hypergeometric multiple summation identities. The package has been developed by Axel Riese, a former member of the RISC Combinatorics group. ...

Authors: Axel Riese
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qZeil

A Mathematica Implementation of q-Analogues of Gosper's and Zeilberger's Algorithm

This package is part of the RISCErgoSum bundle. qZeil is a Mathematica implementation of q-analogues of Gosper’s and Zeilberger’s algorithm for proving and finding indefinite and definite q-hypergeometric summation identities. The package has been developed by Axel Riese, a former ...

Authors: Axel Riese
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RatDiff

A Mathematica Implementation of Mark van Hoeij's Algorithm for Finding Rational Solutions of Linear Difference Equations

RatDiff is a Mathematica implementation of Mark van Hoeij's algorithm for finding rational solutions of linear difference equations. The package has been developed by Axel Riese, a Postdoc of the RISC Combinatorics group during a stay at the University of ...

Authors: Axel Riese
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RLangGFun

A Maple Implementation of the Inverse Schützenberger Methodology

The inverse Schützenberger methodology transforms a rational generating function into a (pseudo-) regular expression for a corresponding regular language, and is based on Soittola's Theorem about the N-rationality of a formal power series. It is implemented in the Maple package ...

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Sigma

A Mathematica Package for Discovering and Proving Multi-Sum Identities

Sigma is a Mathematica package that can handle multi-sums in terms of indefinite nested sums and products. The summation principles of Sigma are: telescoping, creative telescoping and recurrence solving. The underlying machinery of Sigma is based on difference field theory. ...

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Stirling

A Mathematica Package for Computing Recurrence Equations of Sums Involving Stirling Numbers or Eulerian Numbers

This package is part of the RISCErgoSum bundle. The Stirling package provides a command for computing recurrence equations of sums involving Stirling numbers or Eulerian numbers. ...

Authors: Manuel Kauers
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SumCracker

A Mathematica Implementation of several Algorithms for Identities and Inequalities of Special Sequences, including Summation Problems

This package is part of the RISCErgoSum bundle. The SumCracker package contains routines for manipulating a large class of sequences (admissible sequences). It can prove identities and inequalities for these sequences, simplify expressions, evaluate symbolic sums, and solve certain difference ...

Authors: Manuel Kauers
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Zeilberger

A Maxima Implementation of Gosper's and Zeilberger's Algorithm

Zeilberger is an implementatian for the free and open source Maxima computer algebra system of Gosper's and Zeilberger's algorithm for proving and finding indefinite and definite hypergeometric summation identities. The package has been developed by Fabrizio Caruso, a former Ph. ...

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Publications

2019

On the positivity of the Gillis–Reznick–Zeilberger rational function

V. Pillwein

Advances in Applied Mathematics 104, pp. 75 - 84. 2019. ISSN 0196-8858. [url]
[bib]
@article{RISC5813,
author = {V. Pillwein},
title = {{On the positivity of the Gillis–Reznick–Zeilberger rational function}},
language = {english},
abstract = {In this note we provide further evidence for a conjecture of Gillis, Reznick, and Zeilberger on the positivity of the diagonal coefficients of a multivariate rational function. Kauers had proven this conjecture for up to 6 variables using computer algebra. We present a variation of his approach that allows us to prove positivity of the coefficients up to 17 variables using symbolic computation.},
journal = {Advances in Applied Mathematics},
volume = {104},
pages = {75 -- 84},
isbn_issn = { ISSN 0196-8858},
year = {2019},
refereed = {yes},
keywords = {Positivity, Cylindrical decomposition, Rational function, Symbolic summation},
length = {10},
url = {http://www.sciencedirect.com/science/article/pii/S0196885818301179}
}

Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable

J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider

Nucl. Phys. B 939, pp. 253-291. 2019. ISSN 0550-3213. arXiv:1810.12261 [hep-ph]. [url]
[bib]
@article{RISC5795,
author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},
title = {{Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable}},
language = {english},
journal = {Nucl. Phys. B},
volume = {939},
pages = {253--291},
isbn_issn = {ISSN 0550-3213},
year = {2019},
note = {arXiv:1810.12261 [hep-ph]},
refereed = {yes},
length = {39},
url = {https://www.sciencedirect.com/science/article/pii/S055032131830350X?via%3Dihub}
}

Numerical Implementation of Harmonic Polylogarithms to Weight w = 8

J. Ablinger, J. Blümlein, M. Round, C. Schneider

To appear in Comput. Phys. Comm., pp. 1-19. 2019. ISSN 0010-4655. arXiv:1809.07084 [hep-ph]. [url]
[bib]
@article{RISC5751,
author = {J. Ablinger and J. Blümlein and M. Round and C. Schneider},
title = {{Numerical Implementation of Harmonic Polylogarithms to Weight w = 8}},
language = {english},
journal = {To appear in Comput. Phys. Comm.},
pages = {1--19},
isbn_issn = {ISSN 0010-4655},
year = {2019},
note = {arXiv:1809.07084 [hep-ph]},
refereed = {yes},
length = {19},
url = {https://arxiv.org/pdf/1809.07084.pdf}
}

Towards a symbolic summation theory for unspecified sequences

P. Paule, C. Schneider

In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, J. Blümlein, P. Paule, C. Schneider (ed.), Texts and Monographs in Symbolic Computation , pp. 351-390. 2019. Springer, ISBN 978-3-030-04479-4. arXiv:1809.06578 [cs.SC]. [url]
[bib]
@incollection{RISC5750,
author = {P. Paule and C. Schneider},
title = {{Towards a symbolic summation theory for unspecified sequences}},
booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},
language = {english},
series = {Texts and Monographs in Symbolic Computation},
pages = {351--390},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-04479-4},
year = {2019},
note = {arXiv:1809.06578 [cs.SC]},
editor = {J. Blümlein and P. Paule and C. Schneider},
refereed = {yes},
length = {40},
url = {https://arxiv.org/abs/1809.06578}
}

A Family of Congruences for Rogers-Ramanujan Subpartitions

Nicolas Allen Smoot

Journal of Number Theory 196, pp. 35-60. March 2019. ISSN 0022-314X. [pdf]
[bib]
@article{RISC5809,
author = {Nicolas Allen Smoot},
title = {{A Family of Congruences for Rogers--Ramanujan Subpartitions}},
language = {english},
abstract = {In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, A1(m), which counted subpartitions with a structure related to the Rogers–Ramanujan identities. They conjectured the existence of an infinite class of congruences for A1(m), modulo powers of 5. We give an explicit form of this conjecture, and prove it for all powers of 5.},
journal = {Journal of Number Theory},
volume = {196},
pages = {35--60},
isbn_issn = {ISSN 0022-314X},
year = {2019},
month = {March},
refereed = {yes},
keywords = {Integer partitions, Partition congruences, Rogers--Ramanujan identities, Ramanujan--Kolberg identities, Modular functions},
sponsor = {FWF: W1214-N15},
length = {26}
}

2018

An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences

J. Ablinger

In: Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018, J. Blümlein and P. Marquard (ed.), PoS(LL2018) , pp. 1-10. 2018. ISSN 1824-8039. [url]
[bib]
@inproceedings{RISC5789,
author = {J. Ablinger},
title = {{An Improved Method to Compute the Inverse Mellin Transform of Holonomic Sequences}},
booktitle = {{Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018}},
language = {english},
series = {PoS(LL2018)},
pages = {1--10},
isbn_issn = {ISSN 1824-8039},
year = {2018},
editor = {J. Blümlein and P. Marquard},
refereed = {yes},
length = {10},
url = {https://pos.sissa.it/303/063/pdf}
}

Polynomial Identities Implying Capparelli's Partition Theorems

Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (submitted), pp. -. 2018. N/A. [url]
[bib]
@article{RISC5790,
author = {Ali Kemal Uncu and Alexander Berkovich},
title = {{Polynomial Identities Implying Capparelli's Partition Theorems }},
language = {english},
journal = {ArXiv e-prints (submitted)},
pages = {--},
isbn_issn = {N/A},
year = {2018},
refereed = {yes},
length = {21},
url = {https://arxiv.org/pdf/1807.10974.pdf}
}

Elementary Polynomial Identities Involving q-Trinomial Coefficients

Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (submitted), pp. -. 2018. N/A. [url]
[bib]
@article{RISC5791,
author = {Ali Kemal Uncu and Alexander Berkovich},
title = {{Elementary Polynomial Identities Involving q-Trinomial Coefficients }},
language = {english},
journal = {ArXiv e-prints (submitted)},
pages = {--},
isbn_issn = {N/A},
year = {2018},
refereed = {yes},
length = {0},
url = {https://arxiv.org/abs/1810.06497}
}

Refined q-Trinomial Coefficients and Two Infinite Hierarcies of q-Series Identities

Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (submitted), pp. 1-10. 2018. N/A. [url]
[bib]
@article{RISC5801,
author = {Ali Kemal Uncu and Alexander Berkovich},
title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarcies of q-Series Identities }},
language = {english},
abstract = {We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarcies contains an identity which is equivalent to Capparelli's first Partition Theorem. },
journal = {ArXiv e-prints (submitted)},
pages = {1--10},
isbn_issn = {N/A},
year = {2018},
refereed = {yes},
length = {10},
url = {https://arxiv.org/abs/1810.12048}
}

Dancing Samba with Ramanujan Partition Congruences

Ralf Hemmecke

Journal of Symbolic Compuation 84, pp. 14-24. 2018. ISSN 0747-7171. [url]
[bib]
@article{RISC5703,
author = {Ralf Hemmecke},
title = {{Dancing Samba with Ramanujan Partition Congruences}},
language = {english},
abstract = {The article presents an algorithm to compute a $C[t]$-module basis $G$ for a given subalgebra $A$ over a polynomial ring $R=C[x]$ with a Euclidean domain $C$ as the domain of coefficients and $t$ a given element of $A$. The reduction modulo $G$ allows a subalgebra membership test. The algorithm also works for more general rings $R$, in particular for a ring $R\subset C((q))$ with the property that $f\in R$ is zero if and only if the order of $f$ is positive. As an application, we algorithmically derive an explicit identity (in terms of quotients of Dedekind $\eta$-functions and Klein's $j$-invariant) that shows that $p(11n+6)$ is divisible by 11 for every natural number $n$ where $p(n)$ denotes the number of partitions of $n$.},
journal = {Journal of Symbolic Compuation},
volume = {84},
pages = {14--24},
isbn_issn = {ISSN 0747-7171},
year = {2018},
refereed = {yes},
keywords = {Partition identities, Number theoretic algorithm, Subalgebra basis},
length = {11},
url = {http://www.sciencedirect.com/science/article/pii/S0747717117300147}
}

Algorithmic Arithmetics with DD-Finite Functions

Jiménez-Pastor Antonio, Pillwein Veronika

In: Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation, Arreche Carlos (ed.), ISSAC '18 , pp. 231-237. 2018. ACM, New York, NY, USA, ISBN 978-1-4503-5550-6. [url]
[bib]
@inproceedings{RISC5730,
author = {Jiménez-Pastor Antonio and Pillwein Veronika},
title = {{Algorithmic Arithmetics with DD-Finite Functions}},
booktitle = {{Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation}},
language = {english},
series = {ISSAC '18},
pages = {231--237},
publisher = {ACM},
address = {New York, NY, USA},
isbn_issn = {ISBN 978-1-4503-5550-6},
year = {2018},
editor = {Arreche Carlos},
refereed = {yes},
keywords = {algorithms, closure properties, holonomic functions},
length = {7},
url = {http://doi.acm.org/10.1145/3208976.3209009}
}

A Computable Extension for Holonomic Functions: DD-Finite Functions

Jiménez-Pastor Antonio, Pillwein Veronika

Journal of Symbolic Computation, pp. -. 2018. ISSN 0747-7171. In Press. [url]
[bib]
@article{RISC5831,
author = {Jiménez-Pastor Antonio and Pillwein Veronika},
title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},
language = {english},
abstract = {Differentiably finite (D-finite) formal power series form a large class of useful functions for which a variety of symbolic algorithms exists. Among these methods are several closure properties that can be carried out automatically. We introduce a natural extension of these functions to a larger class of computable objects for which we prove closure properties. These are again algorithmic. This extension can be iterated constructively preserving the closure properties},
journal = {Journal of Symbolic Computation},
pages = {--},
isbn_issn = {ISSN 0747-7171},
year = {2018},
note = {In Press},
refereed = {yes},
length = {15},
url = {https://doi.org/10.1016/j.jsc.2018.07.002}
}

The Method of Brackets in Experimental Mathematics

Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

In: Frontiers in Orthogonal Polynomials and q-Series, Xin Li, Zuhair Nashed (ed.), pp. -. 2018. World Scientific Publishing, 978-981-3228-87-0. [url]
[bib]
@incollection{RISC5497,
author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},
title = {{The Method of Brackets in Experimental Mathematics}},
booktitle = {{Frontiers in Orthogonal Polynomials and q-Series}},
language = {english},
pages = {--},
publisher = {World Scientific Publishing},
isbn_issn = {978-981-3228-87-0},
year = {2018},
editor = {Xin Li and Zuhair Nashed},
refereed = {no},
length = {0},
url = {http://www.worldscientific.com/worldscibooks/10.1142/10677}
}

Holonomic Tools for Basic Hypergeometric Functions

Christoph Koutschan, Peter Paule

In: Frontiers of Orthogonal Polynomials and q-Series, Xin Li, Zuhair Nashed (ed.), pp. ?-?. 2018. World Scientific Publishing, ISBN 978-981-3228-87-0. [pdf]
[bib]
@incollection{RISC5246,
author = {Christoph Koutschan and Peter Paule},
title = {{Holonomic Tools for Basic Hypergeometric Functions}},
booktitle = {{Frontiers of Orthogonal Polynomials and q-Series}},
language = {english},
pages = {?--?},
publisher = {World Scientific Publishing},
isbn_issn = {ISBN 978-981-3228-87-0},
year = {2018},
editor = {Xin Li and Zuhair Nashed},
refereed = {no},
length = {19}
}

The Number of Realizations of a Laman Graph

Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes, Josef Schicho

SIAM Journal on Applied Algebra and Geometry 2(1), pp. 94-125. 2018. 2470-6566. [url]
[bib]
@article{RISC5700,
author = {Jose Capco and Matteo Gallet and Georg Grasegger and Christoph Koutschan and Niels Lubbes and Josef Schicho},
title = {{The Number of Realizations of a Laman Graph}},
language = {english},
journal = {SIAM Journal on Applied Algebra and Geometry},
volume = {2},
number = {1},
pages = {94--125},
isbn_issn = {2470-6566},
year = {2018},
refereed = {yes},
length = {32},
url = {https://doi.org/10.1137/17M1118312}
}

The functional equation of Dedekind's $\eta$-function

Tobias Magnusson

June 15 2018. [pdf] [tex]
[bib]
@techreport{RISC5701,
author = {Tobias Magnusson},
title = {{The functional equation of Dedekind's $\eta$-function}},
language = {English},
year = {2018},
month = {June 15},
length = {14},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

Difference equation satisfied by the Stieltjes transform of a sequence

D. Dominici, V. Pillwein

Doctoral Program "Computational Mathematics". Technical report no. DK Report 2018-11, 2018. [url] [pdf]
[bib]
@techreport{RISC5827,
author = {D. Dominici and V. Pillwein},
title = {{Difference equation satisfied by the Stieltjes transform of a sequence}},
language = {english},
number = {DK Report 2018-11},
year = {2018},
institution = {Doctoral Program "Computational Mathematics"},
length = {13},
url = {https://www.dk-compmath.jku.at/publications/dk-reports/2018-12-12/}
}

Positivity of the Gillis-Reznick-Zeilberger rational function

V. Pillwein

Technical report no. 18-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2018. [pdf]
[bib]
@techreport{RISC5612,
author = {V. Pillwein},
title = {{Positivity of the Gillis-Reznick-Zeilberger rational function}},
language = {english},
number = {18-05},
year = {2018},
length = {12},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams

J. Ablinger, J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, C.G. Raab, C.-S. Radu, C. Schneider

J. Math. Phys. 59(062305), pp. 1-55. 2018. ISSN 0022-2488. arXiv:1706.01299 [hep-th]. [url]
[bib]
@article{RISC5456,
author = {J. Ablinger and J. Blümlein and A. De Freitas and M. van Hoeij and E. Imamoglu and C.G. Raab and C.-S. Radu and C. Schneider},
title = {{Iterated Elliptic and Hypergeometric Integrals for Feynman Diagrams}},
language = {english},
journal = {J. Math. Phys.},
volume = {59},
number = {062305},
pages = {1--55},
isbn_issn = {ISSN 0022-2488},
year = {2018},
note = {arXiv:1706.01299 [hep-th]},
refereed = {no},
length = {55},
url = {https://arxiv.org/abs/1706.01299}
}

The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg, Q}^{(3)}$

J. Ablinger, J. Blümlein, A. De Freitas, A. Goedicke, C. Schneider, K. Schönwald

Nucl. Phys. B 932, pp. 129-240. 2018. ISSN 0550-3213. arXiv:1804.02226 [hep-ph]. [url]
[bib]
@article{RISC5618,
author = {J. Ablinger and J. Blümlein and A. De Freitas and A. Goedicke and C. Schneider and K. Schönwald},
title = {{The Two-mass Contribution to the Three-Loop Gluonic Operator Matrix Element $A_{gg,Q}^{(3)}$}},
language = {english},
journal = {Nucl. Phys. B},
volume = {932},
pages = {129--240},
isbn_issn = {ISSN 0550-3213},
year = {2018},
note = {arXiv:1804.02226 [hep-ph]},
refereed = {yes},
length = {112},
url = {https://doi.org/10.1016/j.nuclphysb.2018.04.023}
}

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