# 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

### Asymptotics

#### A Mathematica Package for Computing Asymptotic Series Expansions of Univariate Holonomic Sequences

This package is part of the RISCErgoSum bundle. The Asymptotics package provides a command for computing asymptotic series expansions of solutions of P-finite recurrence equations. ...

Authors: Manuel Kauers

### 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

### Dependencies

#### A Mathematica Package for Computing Algebraic Relations of C-finite Sequences and Multi-Sequences

This package is part of the RISCErgoSum bundle. For any tuple f_1, f_2,..., f_r of sequences, the set of multivariate polynomials p such that p(f1(n),f2(n),...,fr(n))=0 for all points n forms ...

### 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 ...

### 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 ...

### fastZeil

#### The Paule/Schorn Implementation of Gosper’s and Zeilberger’s Algorithms

This package is part of the RISCErgoSum bundle. With Gosper’s algorithm you can find closed forms for indefinite hypergeometric sums. If you do not succeed, then you may use Zeilberger’s algorithm to come up with a recurrence relation for that ...

### GeneratingFunctions

#### A Mathematica Package for Manipulations of Univariate Holonomic Functions and Sequences

This package is part of the RISCErgoSum bundle. GeneratingFunctions is a Mathematica package for manipulations of univariate holonomic functions and sequences. ...

### 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 ...

### 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

### 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. ...

### 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) ...

### math4ti2

#### A Mathematica Interface to 4ti2

math4ti2.m is an interface package, allowing the execution of zsolve of the package 4ti2 from within Mathematica notebooks. The package is written by Ralf Hemmecke and Silviu Radu. Licence This program is free software: you can redistribute it and/or ...

### 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 ...

### MultiIntegrate

#### The MultiIntegrate package allows to compute multi-dimensional integrals over hyperexponential integrands in terms of (generalized) harmonic sums.

The MultiIntegrate package allows to compute multi-dimensional integrals over hyperexponential integrands in terms of (generalized) harmonic sums. This package uses variations and extensions of the multivariate Alkmkvist-Zeilberger algorithm. Registration and Legal Notices The source code for this package is password ...

### 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 ...

### 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

### ore_algebra

#### A Sage Package for doing Computations with Ore Operators

The ore_algebra package provides an implementation of Ore algebras for Sage. The main features for the most common instances include basic arithmetic and actions; gcrd and lclm; D-finite closure properties; natural transformations between related algebras; guessing; desingularization; solvers for polynomials, ...

### 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 ...

### PermGroup

#### A Mathematica Package for Permutation Groups, Group Actions and Polya Theory

PermGroup is a Mathematica package dealing with permutation groups, group actions and Polya theory. The package has been developed by Thomas Bayer, a former student of the RISC Combinatorics group. ...

Authors: Thomas Bayer

### QEta

#### A FriCAS package to compute with Dedekind eta functions

The QEta package is a collection of programs written in the FriCAS computer algebra system that allow to compute with Dedekind eta-functions and related q-series where q=exp(2 π i τ). Furthermore, we provide a number of functions connected to the ...

Authors: Ralf Hemmecke

### qFunctions

#### The qFunctions package is a Mathematica package for q-series and partition theory applications.

The qFunctions package by Jakob Ablinger and Ali K. Uncu is a Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-shift equations and recurrences ...

### qGeneratingFunctions

#### A Mathematica Package for Manipulations of Univariate q-Holonomic Functions and Sequences

This package is part of the RISCErgoSum bundle. The qGeneratingFunctions package provides commands for manipulating q-holonomic sequences and power series. ...

### 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

### 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

RaduRK is a Mathematica implementation of an algorithm developed by Cristian-Silviu Radu. The algorithm takes as input an arithmetic sequence a(n) generated from a large class of q-Pochhammer quotients, together with a given arithmetic progression mn+j, and the level of ...

Authors: Nicolas Smoot

### 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

### 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 ...

### 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. ...

### 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

### 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

### 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. ...

## Publications

[Ablinger]

### Proving Two Conjectural Series for $\zeta(7)$ and Discovering More Series for $\zeta(7)$

#### J. Ablinger

In: Mathematical Aspects of Computer and Information Science, , pp. 42-47. 2020. Springer International Publishing, 978-3-030-43120-4. [url]
@inproceedings{RISC6102,
author = {J. Ablinger},
title = {{Proving Two Conjectural Series for $\zeta(7)$ and Discovering More Series for $\zeta(7)$}},
booktitle = {{Mathematical Aspects of Computer and Information Science}},
language = {english},
pages = {42--47},
publisher = {Springer International Publishing},
isbn_issn = {978-3-030-43120-4},
year = {2020},
editor = {D. Slamanig and E. Tsigaridas and Z. Zafeirakopoulos},
refereed = {yes},
length = {6},
url = {https://arxiv.org/abs/1908.06631v1}
}
[Ablinger]

### Subleading logarithmic QED initial state corrections to $e^+e^−\to γ^⁎/Z^{0⁎}$ to $O(\alpha^6L^5)$

#### J. Ablinger, J. Blümlein, A. De Freitas, K. Schönwald

Nuclear Physics B 955, pp. 115045-115045. 2020. ISSN 0550-3213. [url]
@article{RISC6111,
author = {J. Ablinger and J. Blümlein and A. De Freitas and K. Schönwald},
title = {{Subleading logarithmic QED initial state corrections to $e^+e^−\to γ^⁎/Z^{0⁎}$ to $O(\alpha^6L^5)$}},
language = {english},
journal = {Nuclear Physics B},
volume = {955},
pages = {115045--115045},
isbn_issn = { ISSN 0550-3213},
year = {2020},
refereed = {yes},
length = {0},
url = {http://www.sciencedirect.com/science/article/pii/S0550321320301310}
}
[Goswami]

### On sums of coefficients of polynomials related to the Borwein conjectures

#### Ankush Goswami, Venkata Raghu Tej Pantangi

May 2020. [pdf]
@techreport{RISC6113,
author = {Ankush Goswami and Venkata Raghu Tej Pantangi},
title = {{On sums of coefficients of polynomials related to the Borwein conjectures}},
language = {english},
abstract = {Recently, Li (Int. J. Number Theory 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci. China. Math. 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $p=3, 5$, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J. 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $p=2, 3, 5, 7, 11, 13$ and for $p>15$, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large $n$.},
year = {2020},
month = {May},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Goswami]

### Some formulae for coefficients in restricted $q$-products

#### Ankush Goswami, Venkata Raghu Tej Pantangi

May 2020. [pdf]
@techreport{RISC6114,
author = {Ankush Goswami and Venkata Raghu Tej Pantangi},
title = {{Some formulae for coefficients in restricted $q$-products}},
language = {english},
abstract = {In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result (see Theorem \ref{main00}) holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (Quarterly J. Math. 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.},
year = {2020},
month = {May},
length = {11},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Jimenez Pastor]

### Some structural results on D^n finite functions

#### A. Jimenez-Pastor, V. Pillwein, M.F. Singer

Advances in Applied Mathematics 117, pp. 0-0. June 2020. Elsevier, 0196-8858. [url] [pdf]
@article{RISC6077,
author = {A. Jimenez-Pastor and V. Pillwein and M.F. Singer},
title = {{Some structural results on D^n finite functions}},
language = {english},
abstract = {D-finite (or holonomic) functions satisfy linear differential equations with polynomial coefficients. They form a large class of functions that appear in many applications in Mathematics or Physics. It is well-known that these functions are closed under certain operations and these closure properties can be executed algorithmically. Recently, the notion of D-finite functions has been generalized to differentially definable or Dn-finite functions. Also these functions are closed under operations such as forming (anti)derivative, addition or multiplication and, again, these can be implemented. In this paper we investigate how Dn-finite functions behave under composition and how they are related to algebraic and differentially algebraic functions.},
journal = {Advances in Applied Mathematics},
volume = {117},
pages = {0--0},
publisher = {Elsevier},
isbn_issn = {0196-8858},
year = {2020},
month = {June},
refereed = {yes},
length = {0},
url = {https://doi.org/10.1016/j.aam.2020.102027}
}
[Paule]

### Holonomic Relations for Modular Functions and Forms: First Guess, then Prove

2020. [pdf]
@techreport{RISC6081,
author = {Peter Paule and Silviu Radu},
title = {{Holonomic Relations for Modular Functions and Forms: First Guess, then Prove}},
language = {english},
year = {2020},
length = {46},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Paule]

### An algorithm to prove holonomic differential equations for modular forms

May 2020. [pdf]
@techreport{RISC6108,
author = {Peter Paule and Cristian-Silviu Radu},
title = {{An algorithm to prove holonomic differential equations for modular forms}},
language = {english},
abstract = {Express a modular form $g$ of positive weight locally in terms of a modular function $h$ as$y(h)$, say. Then $y(h)$ as a function in $h$ satisfiesa holonomic differential equation; i.e., one which islinear with coefficients being polynomials in $h$.This fact traces back to Gau{\ss} and has beenpopularized prominently by Zagier. Using holonomicprocedures, computationally it is often straightforwardto derive such differential equations as conjectures.In the spirit of the first guess, then prove'' paradigm,we present a new algorithm to prove such conjectures.},
year = {2020},
month = {May},
keywords = {modular functions, modular forms, holonomic differential equations, holonomic functions and sequences, Fricke-Klein relations, $q$-series, partition congruences},
length = {48},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Pillwein]

### A sequence of polynomials generated by a Kapteyn series of the second kind

#### D. Dominici, V. Pillwein

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, , Texts and Monographs in Symbolic Computuation, in press , pp. ?-?. 2020. Springer, arXiv:1607.05314 [math.CO]. [url] [pdf]
@incollection{RISC6078,
author = {D. Dominici and V. Pillwein},
title = {{A sequence of polynomials generated by a Kapteyn series of the second kind}},
booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},
language = {english},
series = {Texts and Monographs in Symbolic Computuation, in press},
pages = {?--?},
publisher = {Springer},
isbn_issn = {?},
year = {2020},
note = {arXiv:1607.05314 [math.CO]},
editor = {V. Pillwein and C. Schneider},
refereed = {yes},
length = {0},
url = {https://www.dk-compmath.jku.at/publications/dk-reports/2019-05-28/view}
}

### A Reduction Theorem of Certain Relations Modulo p Involving Modular Forms

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, , pp. 1-15. 2020. Springer, 978-3-030-44558-4. [pdf]
@incollection{RISC6110,
title = {{A Reduction Theorem of Certain Relations Modulo p Involving Modular Forms}},
booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},
language = {english},
pages = {1--15},
publisher = {Springer},
isbn_issn = {978-3-030-44558-4},
year = {2020},
editor = {V. Pillwein and C. Schneider},
refereed = {yes},
length = {15}
}
[Schneider]

### Evaluation of binomial double sums involving absolute values

#### C. Krattenthaler, C. Schneider

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, , Texts and Monographs in Symbolic Computuation, in press , pp. ?-?. 2020. Springer, arXiv:1607.05314 [math.CO]. [url]
@incollection{RISC5970,
author = {C. Krattenthaler and C. Schneider},
title = {{Evaluation of binomial double sums involving absolute values}},
booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},
language = {english},
series = {Texts and Monographs in Symbolic Computuation, in press},
pages = {?--?},
publisher = {Springer},
isbn_issn = {?},
year = {2020},
note = {arXiv:1607.05314 [math.CO]},
editor = {V. Pillwein and C. Schneider},
refereed = {yes},
length = {36},
url = {https://arxiv.org/abs/1607.05314}
}
[Schneider]

### Minimal representations and algebraic relations for single nested products

#### C. Schneider

Programming and Computer Software 46(2), pp. 133-161. 2020. ISSN 1608-3261. arXiv:1911.04837 [cs.SC]. [url]
@article{RISC6002,
author = {C. Schneider},
title = {{Minimal representations and algebraic relations for single nested products}},
language = {english},
journal = {Programming and Computer Software},
volume = {46},
number = {2},
pages = {133--161},
isbn_issn = {ISSN 1608-3261},
year = {2020},
note = {arXiv:1911.04837 [cs.SC]},
refereed = {yes},
length = {29},
url = {https://arxiv.org/abs/1911.04837}
}
[Schneider]

### Three loop QCD corrections to heavy quark form factors

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

In: Proceedings of ACAT 2019, to appear, pp. -. 2020. arXiv:1905.03728 [hep-ph]. [url]
@inproceedings{RISC6004,
author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},
title = {{Three loop QCD corrections to heavy quark form factors}},
booktitle = {{Proceedings of ACAT 2019}},
language = {english},
volume = {to appear},
pages = {--},
isbn_issn = {?},
year = {2020},
note = {arXiv:1905.03728 [hep-ph]},
editor = {?},
refereed = {no},
length = {9},
url = {https://arxiv.org/abs/1905.03728}
}
[Schneider]

### From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms

#### J. Blümlein, A. Maier, P. Marquard, G. Schäfer, C. Schneider

Physics Letters B 801(135157), pp. 1-8. 2020. ISSN 0370-2693. arXiv:1911.04411 [gr-qc]. [url]
@article{RISC6009,
author = {J. Blümlein and A. Maier and P. Marquard and G. Schäfer and C. Schneider},
title = {{From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms}},
language = {english},
journal = {Physics Letters B},
volume = {801},
number = {135157},
pages = {1--8},
isbn_issn = {ISSN 0370-2693},
year = {2020},
note = {arXiv:1911.04411 [gr-qc]},
refereed = {yes},
length = {8},
url = {https://doi.org/10.1016/j.physletb.2019.135157}
}
[Schneider]

### Heavy quark form factors at three loops

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

@inproceedings{RISC6014,
author = {J. Blümlein and P. Marquard and N. Rana and C. Schneider},
title = {{Heavy quark form factors at three loops}},
language = {english},
pages = {1--7},
isbn_issn = {ISSN 1824-8039},
year = {2020},
editor = {D. Kosower and M. Cacciari},
refereed = {yes},
length = {7},
url = {https://doi.org/10.22323/1.375.0013 }
}
[Schneider]

### A refined machinery to calculate large moments from coupled systems of linear differential equations

#### Johannes Blümlein, Peter Marquard, Carsten Schneider

In: 14th International Symposium on Radiative Corrections (RADCOR2019), , POS(RADCOR2019)078 , pp. 1-13. 2020. ISSN 1824-8039. arXiv:1912.04390 [cs.SC]. [url]
@inproceedings{RISC6015,
author = {Johannes Blümlein and Peter Marquard and Carsten Schneider},
title = {{A refined machinery to calculate large moments from coupled systems of linear differential equations}},
language = {english},
pages = {1--13},
isbn_issn = {ISSN 1824-8039},
year = {2020},
note = {arXiv:1912.04390 [cs.SC]},
editor = {D. Kosower and M. Cacciari},
refereed = {yes},
length = {13},
url = {https://doi.org/10.22323/1.375.0078}
}
[Schneider]

### The Polarized Three-Loop Anomalous Dimensions from a Massive Calculation

#### A. Behring, J. Blümlein, A. De Freitas, A. Goedicke, S. Klein, A. van Manteuffel, C. Schneider, K. Schönwald

In: 14th International Symposium on Radiative Corrections (RADCOR2019), , POS(RADCOR2019)047 arXiv:1911.06189 [hep-ph], pp. 1-10. 2020. ISSN 1824-8039. [url]
@inproceedings{RISC6016,
author = {A. Behring and J. Blümlein and A. De Freitas and A. Goedicke and S. Klein and A. van Manteuffel and C. Schneider and K. Schönwald},
title = {{The Polarized Three-Loop Anomalous Dimensions from a Massive Calculation}},
language = {english},
number = {arXiv:1911.06189 [hep-ph]},
pages = {1--10},
isbn_issn = {ISSN 1824-8039},
year = {2020},
editor = {D. Kosower and M. Cacciari},
refereed = {yes},
length = {10},
url = {https://doi.org/10.22323/1.375.0047}
}
[Schneider]

### The three-loop polarized pure singlet operator matrix element with two different masses

#### J. Ablinger, J. Blümlein, A. De Freitas, M. Saragnese, C. Schneider, K. Schönwald

Nuclear Physics B 952(114916), pp. 1-18. 2020. ISSN 0550-3213. arXiv:1911.11630 [hep-ph]. [url]
@article{RISC6017,
author = {J. Ablinger and J. Blümlein and A. De Freitas and M. Saragnese and C. Schneider and K. Schönwald},
title = {{The three-loop polarized pure singlet operator matrix element with two different masses}},
language = {english},
journal = {Nuclear Physics B },
volume = {952},
number = {114916},
pages = {1--18},
isbn_issn = {ISSN 0550-3213},
year = {2020},
note = {arXiv:1911.11630 [hep-ph]},
refereed = {yes},
length = {18},
url = {https://doi.org/10.1016/j.nuclphysb.2020.114916}
}
[Schneider]

### Three loop heavy quark form factors and their asymptotic behavior

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

In: To appear in Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018, , pp. ?-?. 2020. arXiv:1906.05829 [hep-ph]. [url]
@inproceedings{RISC6024,
author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},
title = {{Three loop heavy quark form factors and their asymptotic behavior}},
booktitle = {{To appear in Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018}},
language = {english},
pages = {?--?},
isbn_issn = {?},
year = {2020},
note = {arXiv:1906.05829 [hep-ph]},
editor = {?},
refereed = {no},
length = {8},
url = {https://arxiv.org/abs/1906.05829}
}
[Schneider]

### The three-loop single mass polarized pure singlet operator matrix element

#### J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schönwald

Nuclear Physics B 953(114945), pp. 1-25. 2020. ISSN 0550-3213. arXiv:1912.02536 [hep-ph]. [url]
@article{RISC6075,
author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schönwald},
title = {{The three-loop single mass polarized pure singlet operator matrix element}},
language = {english},
journal = {Nuclear Physics B},
volume = {953},
number = {114945},
pages = {1--25},
isbn_issn = {ISSN 0550-3213},
year = {2020},
note = {arXiv:1912.02536 [hep-ph]},
refereed = {yes},
length = {25},
url = {https://doi.org/10.1016/j.nuclphysb.2020.114945}
}
[Schneider]

### The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra

#### C. Schneider

arXiv. Technical report, 2020. arXiv:2003.01921 [math.CO]. [url]
@techreport{RISC6083,
author = {C. Schneider},
title = {{The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra}},
language = {english},
year = {2020},
note = {arXiv:2003.01921 [math.CO]},
institution = {arXiv},
length = {11},
url = {https://arxiv.org/abs/2003.01921}
}