# 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 ...

### DrawFunDoms

#### Drawing Fundamental Domains

DrawFunDoms.m is a Mathematica package for drawing fundamental domains for congruence subgroups in the modular group SL2(ℤ). It was written by Paul Kainberger as part of his master’s thesis under supervision of Univ.-Prof. Dr. Peter Paule at the ...

Authors:

### 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 ...

### EvaluateMultiSums

#### A Mathematica package to evaluate multi-sums

A Mathematica package based on Sigma that tries to evaluate automatically multi-sums to expressions in terms of indefinite nested sums defined over (q-)hypergeometric products. ...

### 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

### PLDESolver

#### The PLDESolver package is a Mathematica package to find solutions of parameterized linear difference equations in difference rings.

The PLDESolver package by Jakob Ablinger and Carsten Schneider is a Mathematica package that allows to compute solutions of non-degenerated linear difference operators in difference rings with zero-divisors by reducing it to finding solutions in difference rings that are integral ...

### 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]

### Extensions of the AZ-algorithm and the Package MultiIntegrate

#### J. Ablinger

Technical report no. 21-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2021. [doi] [pdf]
@techreport{RISC6272,
author = {J. Ablinger},
title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},
language = {english},
abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate,can be considered as an enhanced implementation of the (continuous) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},
number = {21-02},
year = {2021},
month = {January},
keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},
length = {25},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Hemmecke]

### Construction of Modular Function Bases for $Gamma_0(121)$ related to $p(11n+6)$

#### Ralf Hemmecke, Peter Paule, Silviu Radu

Integral Transforms and Special Functions 32(5-8), pp. 512-527. 2021. Taylor & Francis, 1065-2469. [doi]
@article{RISC6342,
author = {Ralf Hemmecke and Peter Paule and Silviu Radu},
title = {{Construction of Modular Function Bases for $Gamma_0(121)$ related to $p(11n+6)$}},
language = {english},
abstract = {Motivated by arithmetic properties of partitionnumbers $p(n)$, our goal is to find algorithmicallya Ramanujan type identity of the form$sum_{n=0}^{infty}p(11n+6)q^n=R$, where $R$ is apolynomial in products of the form$e_alpha:=prod_{n=1}^{infty}(1-q^{11^alpha n})$with $alpha=0,1,2$. To this end we multiply theleft side by an appropriate factor such the resultis a modular function for $Gamma_0(121)$ havingonly poles at infinity. It turns out thatpolynomials in the $e_alpha$ do not generate thefull space of such functions, so we were led tomodify our goal. More concretely, we give threedifferent ways to construct the space of modularfunctions for $Gamma_0(121)$ having only poles atinfinity. This in turn leads to three differentrepresentations of $R$ not solely in terms of the$e_alpha$ but, for example, by using as generatorsalso other functions like the modular invariant $j$.},
journal = {Integral Transforms and Special Functions},
volume = {32},
number = {5-8},
pages = {512--527},
publisher = {Taylor & Francis},
isbn_issn = {1065-2469},
year = {2021},
refereed = {yes},
keywords = {Ramanujan identities, bases for modular functions, integral bases},
length = {16},
url = {https://doi.org/10.1080/10652469.2020.1806261}
}
[Jimenez Pastor]

### On C2-Finite Sequences

#### Antonio Jiménez-Pastor, Philipp Nuspl, Veronika Pillwein

In: Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation, , ISSAC '21 , pp. 217-224. 2021. Association for Computing Machinery, New York, NY, USA, ISBN 9781450383820. [doi]
@inproceedings{RISC6348,
author = {Antonio Jiménez-Pastor and Philipp Nuspl and Veronika Pillwein},
title = {{On C2-Finite Sequences}},
booktitle = {{Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation}},
language = {english},
abstract = {Holonomic sequences are widely studied as many objects interesting to mathematiciansand computer scientists are in this class. In the univariate case, these are the sequencessatisfying linear recurrences with polynomial coefficients and also referred to asD-finite sequences. A subclass are C-finite sequences satisfying a linear recurrencewith constant coefficients.We investigate the set of sequences which satisfy linearrecurrence equations with coefficients that are C-finite sequences. These sequencesare a natural generalization of holonomic sequences. In this paper, we show that C2-finitesequences form a difference ring and provide methods to compute in this ring.},
series = {ISSAC '21},
pages = {217--224},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
isbn_issn = {ISBN 9781450383820},
year = {2021},
editor = {Frédéric Chyzak and George Labahn},
refereed = {yes},
keywords = {holonomic sequences, algorithms, closure properties, difference equations},
length = {8},
url = {https://doi.org/10.1145/3452143.3465529}
}
[Paule]

### An Invitation to Analytic Combinatorics

#### Peter Paule (ed.), Stephen Melczer

Texts and Monographs in Symbolic Computation 1st edition, 2021. Springer, 978-3-030-67080-1.
@book{RISC6277,
author = {Peter Paule (ed.) and Stephen Melczer},
title = {{An Invitation to Analytic Combinatorics}},
language = {english},
series = {Texts and Monographs in Symbolic Computation},
publisher = {Springer},
isbn_issn = {978-3-030-67080-1},
year = {2021},
edition = {1st},
translation = {0},
length = {405}
}
[Paule]

### Contiguous Relations and Creative Telescoping

#### Peter Paule

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, , Texts and Monographs in Symbolic Computation , pp. -. 2021. Springer, ISBN 978-3-030-80218-9. To appear. [pdf]
@incollection{RISC6366,
author = {Peter Paule},
title = {{Contiguous Relations and Creative Telescoping}},
booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},
language = {english},
abstract = {This article presents an algorithmic theory of contiguous relations.Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non- terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4F3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.},
series = {Texts and Monographs in Symbolic Computation},
pages = {--},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-80218-9},
year = {2021},
note = {To appear},
editor = {J. Bluemlein and C. Schneider},
refereed = {yes},
length = {61}
}
[Paule]

### MacMahon's partition analysis XIII: Schmidt type partitions and modular forms

#### George E. Andrews and Peter Paule

Journal of Number Theory, pp. -. 2021. Elsevier, ISSN 0022-314X. [doi] [pdf]
@article{RISC6373,
author = {George E. Andrews and Peter Paule },
title = {{MacMahon's partition analysis XIII: Schmidt type partitions and modular forms}},
language = {english},
abstract = {In 1999, Frank Schmidt noted that the number of partitionsof integers with distinct parts in which the first, third,fifth, etc., summands add to $n$ is equal to $p(n)$, thenumber of partitions of $n$. The object of this paper isto provide a context for this result which leads directlyto many other theorems of this nature and which can beviewed as a continuation of our work on elongatedpartition diamonds. Again generating functions areinfinite products built by the Dedekind eta functionwhich, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. },
journal = {Journal of Number Theory},
pages = {--},
publisher = {Elsevier},
isbn_issn = {ISSN 0022-314X},
year = {2021},
refereed = {yes},
length = {18},
url = {http://doi.org/10.1016/j.jnt.2021.09.008},
type = {open access}
}
[Schneider]

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

#### C. Schneider

Mathematics in Computer Science 15(4), pp. 577-588. 2021. ISSN 1661-8289. arXiv:2003.01921 [math.CO]. [doi]
@article{RISC6127,
author = {C. Schneider},
title = {{The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra}},
language = {english},
journal = {Mathematics in Computer Science},
volume = {15},
number = {4},
pages = {577--588},
isbn_issn = {ISSN 1661-8289},
year = {2021},
note = {arXiv:2003.01921 [math.CO]},
refereed = {yes},
length = {12},
url = {https://doi.org/10.1007/s11786-020-00494-w}
}
[Schneider]

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

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

In: Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018, , Springer Proceedings in Physics 261, pp. 91-100. 2021. Springer, ISBN 978-981-33-4407-5. arXiv:1906.05829 [hep-ph], https://doi.org/10.1007/978-981-33-4408-2_14. [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 = {{Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018}},
language = {english},
series = {Springer Proceedings in Physics},
volume = {261},
pages = {91--100},
publisher = {Springer},
isbn_issn = {ISBN 978-981-33-4407-5},
year = {2021},
note = {arXiv:1906.05829 [hep-ph], https://doi.org/10.1007/978-981-33-4408-2_14},
editor = {Behera and P.K. and Bhatnagar and V. and Shukla and P. and Sinha and R.},
refereed = {yes},
length = {10},
url = {https://arxiv.org/abs/1906.05829}
}
[Schneider]

### Solving linear difference equations with coefficients in rings with idempotent representations

#### J. Ablinger, C. Schneider

In: Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation (Proc. ISSAC 21), , pp. 27-34. 2021. ISBN 978-1-4503-8382-0/21/06. arXiv:2102.03307 [cs.SC]. [doi]
@inproceedings{RISC6302,
author = {J. Ablinger and C. Schneider},
title = {{Solving linear difference equations with coefficients in rings with idempotent representations}},
booktitle = {{Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation (Proc. ISSAC 21)}},
language = {english},
pages = {27--34},
isbn_issn = {ISBN 978-1-4503-8382-0/21/06},
year = {2021},
note = {arXiv:2102.03307 [cs.SC]},
editor = {Marc Mezzarobba},
refereed = {yes},
length = {8},
url = {https://doi.org/10.1145/3452143.3465535}
}
[Schneider]

### A case study for ζ(4)

In: Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019, , Proceedings in Mathematics & Statistics 373, pp. 421-435. 2021. Springer, ISBN 978-3-030-84304-5. arXiv:2004.08158 [math.NT]. [doi]
@incollection{RISC6210,
author = {Carsten Schneider and Wadim Zudilin},
title = {{A case study for ζ(4)}},
booktitle = {{Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019}},
language = {english},
series = {Proceedings in Mathematics & Statistics},
volume = {373},
pages = {421--435},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-84304-5},
year = {2021},
note = {arXiv:2004.08158 [math.NT]},
editor = {Alin Bostan and Kilian Raschel},
refereed = {yes},
length = {15},
url = {https://www.doi.org/10.1007/978-3-030-84304-5_17}
}
[Schneider]

### On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ∗-field extensions

#### Sergei A. Abramov, Manuel Bronstein, Marko Petkovšek, Carsten Schneider

J. Symb. Comput. 107, pp. 23-66. 2021. ISSN 0747-7171. arXiv:2005.04944 [cs.SC]. [doi]
@article{RISC6224,
author = {Sergei A. Abramov and Manuel Bronstein and Marko Petkovšek and Carsten Schneider},
title = {{On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ∗-field extensions}},
language = {english},
journal = {J. Symb. Comput.},
volume = {107},
pages = {23--66},
isbn_issn = {ISSN 0747-7171},
year = {2021},
note = {arXiv:2005.04944 [cs.SC]},
refereed = {yes},
length = {44},
url = {https://doi.org/10.1016/j.jsc.2021.01.002}
}
[Schneider]

### The Polarized Transition Matrix Element $A_{g, q}(N)$ of the Variable Flavor Number Scheme at $O(alpha_s^3)$

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

Nuclear Physics B 964, pp. 115331-115356. 2021. ISSN 0550-3213. arXiv:2101.05733 [hep-ph]. [doi]
@article{RISC6278,
author = {A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and K. Schönwald and and C. Schneider},
title = {{The Polarized Transition Matrix Element $A_{g,q}(N)$ of the Variable Flavor Number Scheme at $O(alpha_s^3)$}},
language = {english},
journal = {Nuclear Physics B},
volume = {964},
pages = {115331--115356},
isbn_issn = {ISSN 0550-3213},
year = {2021},
note = {arXiv:2101.05733 [hep-ph]},
refereed = {yes},
length = {26},
url = {https://doi.org/10.1016/j.nuclphysb.2021.115331}
}
[Schneider]

### Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

#### C. Schneider

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, , Texts and Monographs in Symbolic Computuation to appear, pp. ?-?. 2021. Springer, arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03. [url]
@incollection{RISC6287,
author = {C. Schneider},
title = {{Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation}},
booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},
language = {english},
series = {Texts and Monographs in Symbolic Computuation},
volume = {to appear},
pages = {?--?},
publisher = {Springer},
isbn_issn = {?},
year = {2021},
note = {arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03},
editor = {J. Blümlein and C. Schneider},
refereed = {yes},
length = {55},
url = {https://arxiv.org/abs/2102.01471}
}
[Schneider]

### Iterated integrals over letters induced by quadratic forms

#### J. Ablinger, J. Blümlein, C. Schneider

Physical Review D 103(9), pp. 096025-096035. 2021. ISSN 2470-0029. arXiv:2103.08330 [hep-th]. [doi]
@article{RISC6294,
author = {J. Ablinger and J. Blümlein and C. Schneider},
title = {{Iterated integrals over letters induced by quadratic forms}},
language = {english},
journal = {Physical Review D },
volume = {103},
number = {9},
pages = {096025--096035},
isbn_issn = {ISSN 2470-0029},
year = {2021},
note = {arXiv:2103.08330 [hep-th]},
refereed = {yes},
length = {11},
url = {https://www.doi.org/10.1103/PhysRevD.103.096025}
}
[Schneider]

### New 2– and 3–loop heavy flavor corrections to unpolarized and polarized deep-inelastic scattering

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

Technical report no. 21-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2021. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6350,
author = {J. Ablinger and J. Blümlein and A. De Freitas and M. Saragnese and C. Schneider and K. Schönwald},
title = {{New 2– and 3–loop heavy flavor corrections to unpolarized and polarized deep-inelastic scattering}},
language = {english},
abstract = {A survey is given on the new 2-- and 3--loop results for the heavy flavor contributions to deep--inelastic scattering in the unpolarized and the polarized case. We also discuss related new mathematical aspectsapplied in these calculations.},
number = {21-14},
year = {2021},
month = {July},
keywords = {deep inelastic scattering, 3-loop Feynman integrals, symbolic summation, large moment method, special functions},
length = {13},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering

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

Physical Review D 104(3), pp. 1-73. 2021. ISSN 2470-0029. arXiv:2105.09572 [hep-ph]. [doi]
@article{RISC6333,
author = {J. Blümlein and A. De Freitas and M. Saragnese and K. Schönwald and C. Schneider},
title = {{The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering}},
language = {english},
abstract = {We compute the logarithmic contributions to the polarized massive Wilson coefficients fordeep-inelastic scattering in the asymptotic region $Q^2gg m^2$ to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the polarized massiveoperator matrix elements needed in the variable flavor number scheme. The calculationis performed in the Larin scheme. For the massive operator matrix elements $A_{qq,Q}^{(3),PS}$ and $A_{qg,Q}^{(3),S}$the complete results are presented. The expressions are given in Mellin-$N$ space andin momentum fraction $z$-space.},
journal = {Physical Review D},
volume = {104},
number = {3},
pages = {1--73},
isbn_issn = {ISSN 2470-0029},
year = {2021},
note = {arXiv:2105.09572 [hep-ph]},
refereed = {yes},
keywords = {logarithmic contributions to the polarized massive Wilson coefficients, symbolic summation, harmonic sums, harmonic polylogarithm},
length = {73},
url = {https://doi.org/10.1103/PhysRevD.104.034030 }
}
[Schneider]

### The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Nucl. Phys. B 971, pp. 1-44. 2021. ISSN 0550-3213. arXiv:2107.06267 [hep-ph]. [doi]
@article{RISC6362,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements}},
language = {english},
abstract = {We calculate the unpolarized and polarized three--loop anomalous dimensions and splitting functions $P_{rm NS}^+, P_{rm NS}^-$ and $P_{rm NS}^{rm s}$ in QCD in the $overline{sf MS}$ scheme by using the traditional method of space--like off shell massless operator matrix elements. This is a gauge--dependent framework. For the first time we also calculate the three--loop anomalous dimensions $P_{rm NS}^{rm pm, tr}$ for transversity directly. We compare our results to the literature. },
journal = {Nucl. Phys. B},
volume = {971},
pages = {1--44},
isbn_issn = {ISSN 0550-3213},
year = {2021},
note = {arXiv:2107.06267 [hep-ph]},
refereed = {yes},
length = {44},
url = {https://doi.org/10.1016/j.nuclphysb.2021.115542}
}
[Schneider]

### The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Technical report no. 21-19 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). November 2021. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6374,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements}},
language = {english},
abstract = {We calculate the polarized three--loop singlet anomalous dimensions and splitting functions in QCD in the M--scheme by using the traditional method of space--like off--shell massless operator matrix elements. This is a gauge--dependent framework. Here one obtains the anomalous dimensions without referring to gravitational currents. We also calculate the non--singlet splitting function $Delta P_{rm qq}^{(2), rm s, NS}$ and compare our results to the literature. },
number = {21-19},
year = {2021},
month = {November},
keywords = {particle physics, solving recurrences, large moment method, harmonic sums},
length = {29},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Uncu]

### qFunctions - A Mathematica package for q-series and partition theory applications

#### J. Ablinger, A. Uncu

Journal of Symbolic Computation 107, pp. 145-166. 2021. ISSN 0747-7171. arXiv:1910.12410. [doi]
@article{RISC6299,
author = {J. Ablinger and A. Uncu},
title = {{qFunctions -- A Mathematica package for q-series and partition theory applications}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {107},
pages = {145--166},
isbn_issn = {ISSN 0747-7171},
year = {2021},
note = {arXiv:1910.12410},
refereed = {yes},
length = {22},
url = {https://doi.org/10.1016/j.jsc.2021.02.003}
}

[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}
}