Members
Jakob Ablinger
Koustav Banerjee
Nikolai Fadeev
Ankush Goswami
Ralf Hemmecke
Antonio Jimenez Pastor
Christoph Koutschan: on leave
Evans Doe Ocansey
Peter Paule
Veronika Pillwein
Cristian-Silviu Radu
Carsten Schneider
Nicolas Smoot
Ali Uncu
Ongoing Projects
SAGEX – Scattering Amplitudes: from Geometry to Experiment
Extension of Algorithms for D-finite functions [DK15]
Computer Algebra and Combinatorial Inequalities [FWF SFB F050-07]
Computer Algebra for Nested Sums and Products [FWF SFB F050-09]
Partition Analysis [F050-06]
Computer Algebra Tools for Special Functions [DK6]
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. ...
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. ...
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 ...
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 ...
HarmonicSums
A Mathematica Package for dealing with Harmonic Sums, Generalized Harmonic Sums and Cyclotomic Sums and their related Integral Representations
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.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
A Mathematica Package providing Basic Algorithms and Visualization Routines related to the Modular Group, e.g. for Drawing the Tessellation of the Upper Half-Plane
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 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 ...
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. ...
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 ...
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. ...
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 ...
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 ...
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 ...
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. ...
Singular.m is an interface package, allowing the execution of Singular functions from Mathematica notebooks, written by Manuel Kauers and Viktor Levandovskyy. ...
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. ...
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 ...
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
2020
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]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
Discovering and Proving Infinite Pochhammer Sum Identities
J. Ablinger
Experimental Mathematics, pp. 1-15. 2019. Taylor & Francis, 10.1080/10586458.2019.1627254. [url]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}
}
Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.
J. Ablinger
arXiv. Technical report, 2019. [url]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}
}
Polynomial Identities Implying Capparelli's Partition Theorems
Ali Kemal Uncu, Alexander Berkovich
Accepted - Journal of Number Theory, pp. -. 2019. N/A. [url]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}
}
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]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}
}
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.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}
}
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.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}
}
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.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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}
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]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}
}