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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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 ...

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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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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
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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 ...

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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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RaduRK

RaduRK: Ramanujan-Kolberg Program

RaduRK is a Mathematica implementation of Cristian-Silviu Radu’s algorithm designed to compute Ramanujan-Kolberg identities. These are identities between the generating functions of certain classes of arithmetic sequences a(n), restricted to an arithmetic progression, and linear Q-combinations of eta quotients. These ...

Authors: Nicolas Smoot
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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

2020

[Schneider]

Minimal representations and algebraic relations for single nested products

C. Schneider

Programming and Computer Software, in press 46(2), pp. ?-?. 2020. ISSN 1608-3261. arXiv:1911.04837 [cs.SC]. [url]
[bib]
@article{RISC6002,
author = {C. Schneider},
title = {{Minimal representations and algebraic relations for single nested products}},
language = {english},
journal = {Programming and Computer Software, in press},
volume = {46},
number = {2},
pages = {?--?},
isbn_issn = {ISSN 1608-3261},
year = {2020},
note = {arXiv:1911.04837 [cs.SC]},
refereed = {yes},
length = {0},
url = {https://arxiv.org/abs/1911.04837}
}

2019

[Ablinger]

Discovering and Proving Infinite Pochhammer Sum Identities

J. Ablinger

Experimental Mathematics, pp. 1-15. 2019. Taylor & Francis, 10.1080/10586458.2019.1627254. [url]
[bib]
@article{RISC5896,
author = {J. Ablinger},
title = {{Discovering and Proving Infinite Pochhammer Sum Identities}},
language = {english},
journal = {Experimental Mathematics},
pages = {1--15},
publisher = {Taylor & Francis},
isbn_issn = {?},
year = {2019},
note = {10.1080/10586458.2019.1627254},
refereed = {yes},
length = {15},
url = {https://doi.org/10.1080/10586458.2019.1627254}
}
[Ablinger]

Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.

J. Ablinger

arXiv. Technical report, 2019. [url]
[bib]
@techreport{RISC5968,
author = {J. Ablinger},
title = {{Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.}},
language = {english},
year = {2019},
institution = {arXiv},
length = {5},
url = {https://arxiv.org/pdf/1908.06631.pdf}
}
[Berkovich]

Polynomial Identities Implying Capparelli's Partition Theorems

Ali Kemal Uncu, Alexander Berkovich

Accepted - Journal of Number Theory, pp. -. 2019. N/A. [url]
[bib]
@article{RISC5790,
author = {Ali Kemal Uncu and Alexander Berkovich},
title = {{Polynomial Identities Implying Capparelli's Partition Theorems }},
language = {english},
journal = {Accepted - Journal of Number Theory},
pages = {--},
isbn_issn = {N/A},
year = {2019},
refereed = {yes},
length = {21},
url = {https://arxiv.org/pdf/1807.10974.pdf}
}
[Berkovich]

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

Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (accepted), pp. 1-10. 2019. N/A. [url]
[bib]
@article{RISC5801,
author = {Ali Kemal Uncu and Alexander Berkovich},
title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarchies 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 hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem. },
journal = {ArXiv e-prints (accepted)},
pages = {1--10},
isbn_issn = {N/A},
year = {2019},
refereed = {yes},
length = {10},
url = {https://arxiv.org/abs/1810.12048}
}
[Goswami]

A q-analogue for Euler’s evaluations of the Riemann zeta function

Ankush Goswami

Research in Number Theory 5:3, pp. 1-11. 2019. Springer, 10.1007.
[bib]
@article{RISC5959,
author = {Ankush Goswami},
title = {{A q-analogue for Euler’s evaluations of the Riemann zeta function}},
language = {english},
journal = {Research in Number Theory},
volume = {5:3},
pages = {1--11},
publisher = {Springer},
isbn_issn = {10.1007},
year = {2019},
refereed = {yes},
length = {11}
}
[Goswami]

A q-analogue for Euler’s $\zeta(6)=\pi^6/6$

Ankush Goswami

In: Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, Springer-Birkhauser (ed.), Proceedings of Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, pp. 1-5. 2019. none.
[bib]
@inproceedings{RISC5960,
author = {Ankush Goswami},
title = {{A q-analogue for Euler’s $\zeta(6)=\pi^6/6$}},
booktitle = {{Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}},
language = {english},
pages = {1--5},
isbn_issn = {none},
year = {2019},
editor = {Springer-Birkhauser},
refereed = {yes},
length = {5},
conferencename = {Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}
}
[Goswami]

Some Problems in Analytic Number Theory

Ankush Goswami

University of Florida. PhD Thesis. 2019. First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan - A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY.
[bib]
@phdthesis{RISC5961,
author = {Ankush Goswami},
title = {{Some Problems in Analytic Number Theory}},
language = {english},
year = {2019},
note = {First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan -- A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY},
translation = {0},
school = {University of Florida},
length = {90}
}
[Hemmecke]

Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$

Ralf Hemmecke, Silviu Radu

Journal of Symbolic Compuation 95, pp. 39-52. 2019. ISSN 0747-7171. Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf. [url]
[bib]
@article{RISC5703,
author = {Ralf Hemmecke and Silviu Radu},
title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},
language = {english},
abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind$\eta$-functions of level $N$, i.e., among the elements of$\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where$1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of$N$.More precisely, we find a finite generating set (which is also aGr\"obner basis of the ideal $\ker\phi$ where\begin{gather*} \phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},
journal = {Journal of Symbolic Compuation},
volume = {95},
pages = {39--52},
isbn_issn = {ISSN 0747-7171},
year = {2019},
note = {Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf},
refereed = {yes},
keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},
length = {14},
url = {https://doi.org/10.1016/j.jsc.2018.10.001}
}
[Hemmecke]

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

Ralh Hemmecke, Peter Paule, Silviu Radu

Technical report no. 19-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. October 2019. [url] [pdf]
[bib]
@techreport{RISC5983,
author = {Ralh Hemmecke and Peter Paule and Silviu Radu},
title = {{Construction of Modular Function Bases for $\Gamma_0(121)$ related to $p(11n+6)$}},
language = {english},
number = {19-10},
year = {2019},
month = {October},
keywords = {Ramanujan identities, bases for modular functions, integral bases},
sponsor = {FWF (SFB F50-06)},
length = {16},
url = {https://risc.jku.at/people/hemmecke/papers/integralbasis/},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Hemmecke]

The Generators of all Polynomial Relations among Jacobi Theta Functions

Ralf Hemmecke, Silviu Radu, Liangjie Ye

In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Johannes Blümlein and Carsten Schneider and Peter Paule (ed.), Texts & Monographs in Symbolic Computation 18-09, pp. 259-268. 2019. Springer International Publishing, Cham, 978-3-030-04479-4. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf. [url]
[bib]
@incollection{RISC5913,
author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},
title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},
booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},
language = {english},
abstract = {In this article, we consider the classical Jacobi theta functions$\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},
series = {Texts & Monographs in Symbolic Computation},
number = {18-09},
pages = {259--268},
publisher = {Springer International Publishing},
address = {Cham},
isbn_issn = {978-3-030-04479-4},
year = {2019},
note = {Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf},
editor = {Johannes Blümlein and Carsten Schneider and Peter Paule},
refereed = {yes},
length = {9},
url = {https://doi.org/10.1007/978-3-030-04480-0_11}
}
[Jimenez Pastor]

A Computable Extension for Holonomic Functions: DD-Finite Functions

Jiménez-Pastor Antonio, Pillwein Veronika

Journal of Symbolic Computation 94, pp. 90-104. September-October 2019. ISSN 0747-7171. [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},
volume = {94},
pages = {90--104},
isbn_issn = {ISSN 0747-7171},
year = {2019},
month = {September-October},
refereed = {yes},
length = {15},
url = {https://doi.org/10.1016/j.jsc.2018.07.002}
}
[Paule]

A Proof of the Weierstrass Gap Theorem not using the Riemann-Roch Formula

Peter Paule, Silviu Radu

Technical report no. 19-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. To appear in the Annals of Combinatorics: special issue dedicated to George E. Andrews at the occasion of his 80th birthday., May 2019. [pdf]
[bib]
@techreport{RISC5928,
author = {Peter Paule and Silviu Radu},
title = {{A Proof of the Weierstrass Gap Theorem not using the Riemann-Roch Formula}},
language = {english},
number = {19-02},
year = {2019},
month = {May},
howpublished = {To appear in the Annals of Combinatorics: special issue dedicated to George E. Andrews at the occasion of his 80th birthday.},
length = {47},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Paule]

A UNIFIED ALGORITHMIC FRAMEWORK FOR RAMANUJAN'S CONGRUENCES MODULO POWERS OF 5, 7, AND 11

Peter Paule, Silviu Radu

Technical report no. 00-00 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. submitted, 2019. [pdf]
[bib]
@techreport{RISC5979,
author = {Peter Paule and Silviu Radu},
title = {{A UNIFIED ALGORITHMIC FRAMEWORK FOR RAMANUJAN'S CONGRUENCES MODULO POWERS OF 5, 7, AND 11}},
language = {english},
number = {00-00},
year = {2019},
howpublished = {submitted},
length = {49},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Pillwein]

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},
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}
}
[Radu]

COMPUTING AN ORDER COMPLETE BASIS FOR M∞(N) AND APPLICATIONS

Mark van Hoeij and Cristian-Silviu Radu

Technical report no. 19-50 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2019. [pdf]
[bib]
@techreport{RISC5946,
author = {Mark van Hoeij and Cristian-Silviu Radu },
title = {{COMPUTING AN ORDER COMPLETE BASIS FOR M∞(N) AND APPLICATIONS}},
language = {english},
number = {19-50},
year = {2019},
length = {6},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Schneider]

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}
}
[Schneider]

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}
}
[Schneider]

Three loop QCD corrections to heavy quark form factors

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

Arxiv. Technical report, 2019. arXiv:1905.03728 [hep-ph]. [url]
[bib]
@techreport{RISC5927,
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}},
language = {english},
year = {2019},
note = {arXiv:1905.03728 [hep-ph]},
institution = {Arxiv},
length = {9},
url = {https://arxiv.org/abs/1905.03728}
}
[Schneider]

Three loop heavy quark form factors and their asymptotic behavior

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

Cornell University, arxive. Technical report, 2019. arXiv:1906.05829 [hep-ph]. [url]
[bib]
@techreport{RISC5937,
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}},
language = {english},
year = {2019},
note = {arXiv:1906.05829 [hep-ph]},
institution = {Cornell University, arxive},
length = {8},
url = {https://arxiv.org/abs/1906.05829}
}

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