# 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) ...

### 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 ...

### 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

### 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

### 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

### On some polynomials and series of Bloch-Polya Type

#### Berkovich A., Uncu A. K.

ArXiv e-prints (to appear in Proc. of AMS ), pp. -. 2018. Preprint.
@article{RISC5557,
author = {Berkovich A. and Uncu A.~K.},
title = {{On some polynomials and series of Bloch-Polya Type}},
language = {english},
journal = {ArXiv e-prints (to appear in Proc. of AMS )},
pages = {--},
isbn_issn = {Preprint},
year = {2018},
refereed = {yes},
keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A17, 05A19, 11B65, 11P81},
length = {0}
}

### Some Elementary Partition Inequalities and Their Implications

#### Berkovich A., Uncu A. K.

ArXiv e-prints (submitted), pp. -. 2018. Preprint.
@article{RISC5558,
author = {Berkovich A. and Uncu A.~K.},
title = {{Some Elementary Partition Inequalities and Their Implications}},
language = {english},
journal = {ArXiv e-prints (submitted)},
pages = {--},
isbn_issn = {Preprint},
year = {2018},
refereed = {yes},
keywords = {Mathematics - Combinatorics, Mathematics - Number Theory, 05A15, 05A17, 05A19, 05A20, 11B65, 11P81, 11P84, 33D15},
length = {0}
}

### Dancing Samba with Ramanujan Partition Congruences

#### Ralf Hemmecke

Journal of Symbolic Compuation 84, pp. 14-24. 2018. ISSN 0747-7171. [url]
@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, , ISSAC '18 , pp. 231-237. 2018. ACM, New York, NY, USA, ISBN 978-1-4503-5550-6. [url]
@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. accepted.
@article{RISC5731,
author = {Jiménez-Pastor Antonio and Pillwein Veronika},
title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},
language = {english},
journal = {Journal of Symbolic Computation},
pages = {--},
isbn_issn = {ISSN 0747-7171},
year = {2018},
note = {accepted},
refereed = {yes},
length = {0}
}

### 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, , pp. -. 2018. World Scientific Publishing, 978-981-3228-87-0. [url]
@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, , pp. ?-?. 2018. World Scientific Publishing, ISBN 978-981-3228-87-0. [pdf]
@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]
@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}
}

### Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems

#### J. Middeke, textbfC. Schneider

To appear in ACM Communications in Computer Algebra, pp. -. 2018. ISSN 1932-2240. Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18).
@article{RISC5736,
author = {J. Middeke and textbfC.~Schneider},
title = {{Towards a Direct Method for Finding Hypergeometric Solutions of Linear First Order Recurrence Systems}},
language = {english},
journal = {To appear in ACM Communications in Computer Algebra},
pages = {--},
isbn_issn = {ISSN 1932-2240},
year = {2018},
note = {Extended abstract of the poster presentation at 43st International Symposium on Symbolic and Algebraic Computation (ISSAC'18)},
refereed = {yes},
length = {0}
}

### 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]
@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]
@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}
}

### Denominator Bounds for Systems of Recurrence Equations using ΠΣ-Extensions

#### J. Middeke, C. Schneider

In: Advances in Computer Algebra. WWCA 2016., , Springer Proceedings in Mathematics & Statistics 226, pp. 149-173. 2018. Springer, ISSN 2194-1009. arXiv:1705.00280 [cs.SC]. [url]
@incollection{RISC5447,
author = {J. Middeke and C. Schneider},
title = {{Denominator Bounds for Systems of Recurrence Equations using ΠΣ-Extensions}},
booktitle = {{ Advances in Computer Algebra. WWCA 2016.}},
language = {english},
series = {Springer Proceedings in Mathematics & Statistics},
volume = {226},
pages = {149--173},
publisher = {Springer},
isbn_issn = {ISSN 2194-1009},
year = {2018},
note = {arXiv:1705.00280 [cs.SC]},
editor = {C. Schneider and E. Zima},
refereed = {yes},
length = {24},
url = {https://arxiv.org/abs/1705.00280}
}

### Representing (q-)hypergeometric products and mixed versions in difference rings

#### E.D. Ocansey, C. Schneider

In: Advances in Computer Algebra. WWCA 2016., , Springer Proceedings in Mathematics & Statistics 226, pp. 175-213. 2018. Springer, ISSN 2194-1009. arXiv:1705.01368 [cs.SC]. [url]
@incollection{RISC5448,
author = {E.D. Ocansey and C. Schneider},
title = {{Representing (q-)hypergeometric products and mixed versions in difference rings}},
booktitle = {{ Advances in Computer Algebra. WWCA 2016.}},
language = {english},
series = {Springer Proceedings in Mathematics & Statistics},
volume = {226},
pages = {175--213},
publisher = {Springer},
isbn_issn = {ISSN 2194-1009},
year = {2018},
note = {arXiv:1705.01368 [cs.SC]},
editor = {C. Schneider and E. Zima},
refereed = {yes},
length = {36},
url = {https://arxiv.org/abs/1705.01368}
}

### The ρ parameter at three loops and elliptic integrals

#### J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, P. Marquard, C. Schneider

In: Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018", 29 April - 4 May 2018, , PoS(LL2018)017 , pp. 1-14. 2018. ISSN 1824-8039. arXiv:1807.05287 [hep-ph]. [url]
@inproceedings{RISC5722,
author = {J. Blümlein and A. De Freitas and M. van Hoeij and E. Imamoglu and P. Marquard and C. Schneider},
title = {{The ρ parameter at three loops and elliptic integrals}},
booktitle = {{Proceedings of "Loops and Legs in Quantum Field Theory - LL 2018", 29 April - 4 May 2018}},
language = {english},
series = {PoS(LL2018)017},
pages = {1--14},
isbn_issn = {ISSN 1824-8039},
year = {2018},
note = {arXiv:1807.05287 [hep-ph]},
editor = {J. Blümlein and P. Marquard},
refereed = {yes},
length = {14},
url = {https://pos.sissa.it/303/017/pdf}
}

### Refined Holonomic Summation Algorithms in Particle Physics

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

In: Advances in Computer Algebra. WWCA 2016., , Springer Proceedings in Mathematics & Statistics 226, pp. 51-91. 2018. Springer, ISSN 2194-1009. arXiv:1706.03677 [cs.SC]. [url]
@incollection{RISC5462,
author = {J. Blümlein and M. Round and C. Schneider},
title = {{Refined Holonomic Summation Algorithms in Particle Physics}},
booktitle = {{ Advances in Computer Algebra. WWCA 2016.}},
language = {english},
series = {Springer Proceedings in Mathematics & Statistics },
volume = {226},
pages = {51--91},
publisher = {Springer},
isbn_issn = {ISSN 2194-1009},
year = {2018},
note = {arXiv:1706.03677 [cs.SC]},
editor = {E. Zima and C. Schneider},
refereed = {yes},
length = {39},
url = {https://arxiv.org/abs/1706.03677}
}

### The two-mass contribution to the three-loop pure singlet operator matrix element

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

Nucl. Phys. B(927), pp. 339-367. 2018. ISSN 0550-3213. arXiv:1711.06717 [hep-ph]. [url]
@article{RISC5504,
author = {J. Ablinger and J. Blümlein and A. De Freitas and C. Schneider and K. Schönwald},
title = {{The two-mass contribution to the three-loop pure singlet operator matrix element}},
language = {english},
journal = {Nucl. Phys. B},
number = {927},
pages = {339--367},
isbn_issn = {ISSN 0550-3213},
year = {2018},
note = {arXiv:1711.06717 [hep-ph]},
refereed = {yes},
length = {29},
url = {https://arxiv.org/abs/1711.06717}
}

### Algebraic independence of sequences generated by (cyclotomic) harmonic sums

#### J. Ablinger, C. Schneider

Annals of Combinatorics 22(2), pp. 213-244. 2018. ISSN: 0218-0006. arXiv:1510.03692 [cs.SC], doi 10.1007/s00026-018-0381-5. [url]
@article{RISC5507,
author = {J. Ablinger and C. Schneider},
title = {{Algebraic independence of sequences generated by (cyclotomic) harmonic sums}},
language = {english},
journal = {Annals of Combinatorics},
volume = {22},
number = {2},
pages = {213--244},
isbn_issn = {ISSN: 0218-0006},
year = {2018},
note = {arXiv:1510.03692 [cs.SC], doi 10.1007/s00026-018-0381-5},
refereed = {yes},
length = {32},
}

### The Heavy Quark Form Factors at Two Loops

#### J. Ablinger, A. Behring, J. Bluemlein, G. Falcioni, A. De Freitas, P. Marquard, N. Rana, C. Schneider

Physical Review D 97(094022), pp. 1-44. 2018. ISSN 1550-2368. arXiv:1712.09889 [hep-ph]. [url]
@article{RISC5522,
author = {J. Ablinger and A. Behring and J. Bluemlein and G. Falcioni and A. De Freitas and P. Marquard and N. Rana and C. Schneider},
title = {{The Heavy Quark Form Factors at Two Loops}},
language = {english},
journal = {Physical Review D},
volume = {97},
number = {094022},
pages = {1--44},
isbn_issn = {ISSN 1550-2368},
year = {2018},
note = {arXiv:1712.09889 [hep-ph]},
refereed = {yes},
length = {131},
url = {https://doi.org/10.1103/PhysRevD.97.094022}
}

### The Variable Flavor Number Scheme at Next-to-Leading Order

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

Physics Letters B 782, pp. 362-366. 2018. ISSN: 0370-2693. arXiv:1804.03129 [hep-ph]. [url]
@article{RISC5619,
author = {J. Blümlein and A. De Freitas and C. Schneider and K. Schönwald},
title = {{The Variable Flavor Number Scheme at Next-to-Leading Order}},
language = {english},
journal = {Physics Letters B},
volume = {782},
pages = {362--366},
isbn_issn = {ISSN: 0370-2693},
year = {2018},
note = {arXiv:1804.03129 [hep-ph]},
refereed = {yes},
length = {10},
url = { https://doi.org/10.1016/j.physletb.2018.05.054}
}

### Heavy Quark Form Factors at Three Loops in the Planar Limit

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

Physics Letters B 782, pp. 528-532. 2018. ISSN: 0370-2693. arXiv:1804.07313 [hep-ph]. [url]
@article{RISC5689,
author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},
title = {{Heavy Quark Form Factors at Three Loops in the Planar Limit}},
language = {english},
journal = {Physics Letters B},
volume = {782},
pages = {528--532},
isbn_issn = {ISSN: 0370-2693},
year = {2018},
note = {arXiv:1804.07313 [hep-ph]},
refereed = {yes},
length = {9},
url = {https://doi.org/10.1016/j.physletb.2018.05.077}
}