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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Paul Kainberger Short description 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. ...

Authors:
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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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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 ...

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PositiveSequence

A Mathematica package for showing positivity of univariate C-finite and holonomic sequences

This package is part of the RISCErgoSum bundle. See Download and Installation. Short Description The PositiveSequence package provides methods to show positivity of C-finite and holonomic sequences. Accompanying files Demo.nb Hints Type ?PositiveSequence for information. The package is developed ...

Authors: Philipp Nuspl
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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 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
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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

2022

[Falkensteiner]

On Formal Power Series Solutions of Algebraic Ordinary Differential Equations

S. Falkensteiner, Yi Zhang, N. Thieu Vo

Mediterranean Journal of Mathematics 19(74), pp. 1-16. March 2022. ISSN 1660-5446. [doi]
[bib]
@article{RISC6490,
author = {S. Falkensteiner and Yi Zhang and N. Thieu Vo},
title = {{On Formal Power Series Solutions of Algebraic Ordinary Differential Equations}},
language = {english},
journal = {Mediterranean Journal of Mathematics},
volume = {19},
number = {74},
pages = {1--16},
isbn_issn = {ISSN 1660-5446},
year = {2022},
month = {March},
refereed = {yes},
keywords = {Formal power series, algebraic differential equation.},
length = {16},
url = {https://doi.org/10.1007/s00009-022-01984-w}
}
[Nuspl]

Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences

P. Nuspl, V. Pillwein

Technical report no. 22-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6479,
author = {P. Nuspl and V. Pillwein},
title = {{Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences}},
language = {english},
abstract = {The class of $C^2$-finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients, i.e., coefficients satisfying a linear recurrence with constant coefficients themselves. Recently, we investigated computational properties of $C^2$-finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring.From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple $C^2$-finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function.},
number = {22-02},
year = {2022},
month = {February},
keywords = {difference equations, holonomic sequences, closure properties, generating functions, algorithms},
length = {16},
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)}
}
[Nuspl]

$C$-finite and $C^2$-finite Sequences in SageMath

P. Nuspl

Technical report no. 22-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6516,
author = {P. Nuspl},
title = {{$C$-finite and $C^2$-finite Sequences in SageMath}},
language = {english},
abstract = {We present the SageMath package rec_sequences which provides methods to compute with sequences satisfying linear recurrences. The package can be used to show inequalities of $C$-finite sequences, i.e., sequences satisfying a linear recurrence relation with constant coefficients. Furthermore, it provides functionality to compute in the $C^2$-finite sequence ring, i.e., to compute closure properties of sequences satisfying a linear recurrence with $C$-finite coefficients.},
number = {22-06},
year = {2022},
month = {June},
keywords = {Difference equations, Closure properties, Inequalities},
length = {4},
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)}
}
[Nuspl]

A comparison of algorithms for proving positivity of linearly recurrent sequences

P. Nuspl, V. Pillwein

Technical report no. 22-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6514,
author = {P. Nuspl and V. Pillwein},
title = {{A comparison of algorithms for proving positivity of linearly recurrent sequences}},
language = {english},
abstract = {Deciding positivity for recursively defined sequences based on only the recursive description as input is usually a non-trivial task. Even in the case of $C$-finite sequences, i.e., sequences satisfying a linear recurrence with constant coefficients, this is only known to be decidable for orders up to five. In this paper, we discuss several methods for proving positivity of $C$-finite sequences and compare their effectiveness on input from the Online Encyclopedia of Integer Sequences (OEIS).},
number = {22-05},
year = {2022},
month = {May},
keywords = {Difference equations Inequalities Holonomic sequences},
length = {17},
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)}
}
[Nuspl]

Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences

P. Nuspl, V. Pillwein

In: ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation, Marc Moreno Maza, Lihong Zhi (ed.), pp. 45-53. 2022. Association for Computing Machinery, ISBN: 978-1-4503-8688-3. [doi]
[bib]
@inproceedings{RISC6520,
author = {P. Nuspl and V. Pillwein},
title = {{Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences}},
booktitle = {{ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation}},
language = {english},
pages = {45--53},
publisher = {Association for Computing Machinery},
isbn_issn = {ISBN: 978-1-4503-8688-3},
year = {2022},
editor = {Marc Moreno Maza and Lihong Zhi},
refereed = {yes},
length = {9},
url = {https://doi.org/10.1145/3476446.3536174}
}
[Schneider]

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

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

Journal of High Energy Physics 2022(193), pp. 0-32. 2022. ISSN 1029-8479 . arXiv:2111.12401 [hep-ph]. [doi]
[bib]
@article{RISC6435,
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. },
journal = {Journal of High Energy Physics},
volume = {2022},
number = {193},
pages = {0--32},
isbn_issn = {ISSN 1029-8479 },
year = {2022},
note = {arXiv:2111.12401 [hep-ph]},
refereed = {yes},
keywords = {particle physics, solving recurrences, large moment method, harmonic sums},
length = {33},
url = {https://doi.org/10.1007/JHEP01(2022)193}
}
[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

SciPost Phys. Proc.(8), pp. 137.1-137.15. 2022. ISSN 2666-4003. DIS2021, arXiv:2107.09350 [hep-ph]. [doi]
[bib]
@article{RISC6497,
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.},
journal = {SciPost Phys. Proc.},
number = {8},
pages = {137.1--137.15},
isbn_issn = {ISSN 2666-4003},
year = {2022},
note = {DIS2021, arXiv:2107.09350 [hep-ph]},
refereed = {yes},
length = {15},
url = {https://www.doi.org/10.21468/SciPostPhysProc.8.137}
}
[Schneider]

The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

J. Blümlein, C. Schneider

J. Phys. A: Math. Theor. to appear, pp. ?-?. 2022. IOP Publishing Ltd, ISSN: 1751-8121. arXiv:2203.13015 [hep-th]. [doi]
[bib]
@article{RISC6523,
author = {J. Blümlein and C. Schneider},
title = {{The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals}},
language = {english},
abstract = {The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.},
journal = { J. Phys. A: Math. Theor.},
volume = {to appear},
pages = {?--?},
publisher = { IOP Publishing Ltd},
isbn_issn = {ISSN: 1751-8121},
year = {2022},
note = {arXiv:2203.13015 [hep-th]},
refereed = {yes},
keywords = {Feynman integrals, computer algebra, special functions, linear differential equations, linear difference integrals},
length = {0},
url = { https://doi.org/10.1088/1751-8121/ac8086}
}
[Schneider]

The 3-loop anomalous dimensions from off-shell operator matrix elements

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

Technical report no. 22-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6525,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The 3-loop anomalous dimensions from off-shell operator matrix elements}},
language = {english},
abstract = {We report on the calculation of the three--loop polarized and unpolarized flavor non--singlet and the polarized singlet anomalous dimensions using massless off--shell operator matrix elements in a gauge--variant framework. We also reconsider the unpolarized two--loop singlet anomalous dimensions and correct errors in the foregoing literature.},
number = {22-09},
year = {2022},
month = {July},
keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},
length = {12},
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]

Computer Algebra and Hypergeometric Structures for Feynman Integrals

J. Bluemlein, M. Saragnese, C. Schneider

In: To appear in Proc. of Loops and Legs in Quantum Field Theory - LL 2022, P. Marquard, M. Steinhauser (ed.)PoS(LL2022)041, pp. 1-11. 2022. ISSN 1824-8039. arXiv:2207.08524 [math-ph]. [doi]
[bib]
@inproceedings{RISC6528,
author = {J. Bluemlein and M. Saragnese and C. Schneider},
title = {{Computer Algebra and Hypergeometric Structures for Feynman Integrals}},
booktitle = {{To appear in Proc. of Loops and Legs in Quantum Field Theory - LL 2022}},
language = {english},
abstract = {We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by hypergeometric products and more generally by indefinite nested sums defined over such products. Special cases are hypergeometric structures such as Appell-functions or generalizations of them that arise frequently when dealing with parameter Feynman integrals.},
volume = {PoS(LL2022)041},
pages = {1--11},
isbn_issn = {ISSN 1824-8039},
year = {2022},
note = { arXiv:2207.08524 [math-ph]},
editor = {P. Marquard and M. Steinhauser},
refereed = {no},
keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},
length = {11},
url = {https://doi.org/10.48550/arXiv.2207.08524}
}
[Schneider]

The SAGEX Review on Scattering Amplitudes

G. Travaglini, A. Brandhuber, P. Dorey, T. McLoughlin, S. Abreu, Z. Bern, N. E. J. Bjerrum-Bohr, J. Bluemlein, R. Britto, J. J. M. Carrasco, D. Chicherin, M. Chiodaroli, P. H. Damgaard, V. Del Duca, L. J. Dixon, D. Dorigoni, C. Duhr, Y. Geyer, M. B. Green, E. Herrmann, P. Heslop, H. Johansson, G. P. Korchemsky, D. A. Kosower, L. Mason, R. Monteiro, D. O'Connell, G. Papathanasiou, L. Plante, J. Plefka, A. Puhm, A.-M. Raclariu, R. Roiban, C. Schneider, J. Trnka, P. Vanhove, C. Wen, C. D. White

J. Phys. A: Math. Theor. to appear, pp. ?-?. 2022. arxiv.2203.13011 [hep-th], ISSN: 1751-8121. [url]
[bib]
@article{RISC6529,
author = {G. Travaglini and A. Brandhuber and P. Dorey and T. McLoughlin and S. Abreu and Z. Bern and N. E. J. Bjerrum-Bohr and J. Bluemlein and R. Britto and J. J. M. Carrasco and D. Chicherin and M. Chiodaroli and P. H. Damgaard and V. Del Duca and L. J. Dixon and D. Dorigoni and C. Duhr and Y. Geyer and M. B. Green and E. Herrmann and P. Heslop and H. Johansson and G. P. Korchemsky and D. A. Kosower and L. Mason and R. Monteiro and D. O'Connell and G. Papathanasiou and L. Plante and J. Plefka and A. Puhm and A.-M. Raclariu and R. Roiban and C. Schneider and J. Trnka and P. Vanhove and C. Wen and C. D. White},
title = {{The SAGEX Review on Scattering Amplitudes}},
language = {english},
journal = {J. Phys. A: Math. Theor.},
volume = {to appear},
pages = {?--?},
isbn_issn = {ISSN: 1751-8121},
year = {2022},
refereed = {yes},
institution = {arxiv.2203.13011 [hep-th]},
keywords = {High Energy Physics - Theory (hep-th), General Relativity and Quantum Cosmology (gr-qc), High Energy Physics - Experiment (hep-ex), High Energy Physics - Phenomenology (hep-ph), FOS: Physical sciences},
length = {0},
url = {https://arxiv.org/abs/2203.13011}
}

2021

[Ablinger]

Extensions of the AZ-algorithm and the Package MultiIntegrate

J. Ablinger

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein, C. Schneider (ed.), Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria) , pp. 35-61. 2021. Springer, ISBN 978-3-030-80218-9. arXiv:2101.11385 [cs.SC]. [doi]
[bib]
@incollection{RISC6408,
author = {J. Ablinger},
title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},
booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},
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.},
series = {Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria)},
pages = {35--61},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-80218-9},
year = {2021},
note = {arXiv:2101.11385 [cs.SC]},
editor = {J. Blümlein and C. Schneider},
refereed = {yes},
keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},
length = {27},
url = {https://doi.org/10.1007/978-3-030-80219-6_2}
}
[Ablinger]

Extensions of the AZ-algorithm and the Package MultiIntegrate

J. Ablinger

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein, C. Schneider (ed.), Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria) , pp. 35-61. 2021. Springer, ISBN 978-3-030-80218-9. arXiv:2101.11385 [cs.SC]. [doi]
[bib]
@incollection{RISC6409,
author = {J. Ablinger},
title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},
booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},
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.},
series = {Texts & Monographs in Symbolic Computation (A Series of the Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria)},
pages = {35--61},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-80218-9},
year = {2021},
note = {arXiv:2101.11385 [cs.SC]},
editor = {J. Blümlein and C. Schneider},
refereed = {yes},
keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},
length = {27},
url = {https://doi.org/10.1007/978-3-030-80219-6_2}
}
[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]
[bib]
@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},
sponsor = {FWF (SFB F50-06)},
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, Frédéric Chyzak, George Labahn (ed.), ISSAC '21 , pp. 217-224. 2021. Association for Computing Machinery, New York, NY, USA, ISBN 9781450383820. [doi]
[bib]
@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}
}
[Jimenez Pastor]

An extension of holonomic sequences: C^2-finite sequences

Antonio Jiménez-Pastor, Philipp Nuspl, Veronika Pillwein

Technical report no. 21-20 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). December 2021. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6390,
author = {Antonio Jiménez-Pastor and Philipp Nuspl and Veronika Pillwein},
title = {{An extension of holonomic sequences: C^2-finite sequences}},
language = {english},
abstract = {Holonomic sequences are widely studied as many objects interesting to mathematicians and computer scientists are in this class. In the univariate case, these are the sequences satisfying linear recurrences with polynomial coefficients and also referred to as $D$-finite sequences. A subclass are $C$-finite sequences satisfying a linear recurrence with constant coefficients.We investigate the set of sequences which satisfy linear recurrence equations with coefficients that are $C$-finite sequences. These sequences are a natural generalization of holonomic sequences. In this paper, we show that $C^2$-finite sequences form a difference ring and provide methods to compute in this ring. Furthermore, we provide an analogous construction for $D^2$-finite sequences, i.e., sequences satisfying a linear recurrence with holonomic coefficients. We show that these constructions can be iterated and obtain an increasing chain of difference rings.},
number = {21-20},
year = {2021},
month = {December},
keywords = {Difference equations, holonomic sequences, closure properties, algorithms},
length = {26},
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)}
}
[Jimenez Pastor]

Differentially definable functions: a survey

Jiménez-Pastor Antonio, Pillwein Veronika

In: Proceedings in Applied Mathematics and Mechanics (PAMM), Gesellschaft für Angewandte Mathematik und Mechanik (GAMM) (ed.), Proceedings of Special Issue: 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)21/1, pp. e202100178-e2021001. 2021. [doi]
[bib]
@inproceedings{RISC6410,
author = {Jiménez-Pastor Antonio and Pillwein Veronika},
title = {{Differentially definable functions: a survey}},
booktitle = {{Proceedings in Applied Mathematics and Mechanics (PAMM)}},
language = {english},
abstract = {Abstract Most widely used special functions, such as orthogonal polynomials, Bessel functions, Airy functions, etc., are defined as solutions to differential equations with polynomial coefficients. This class of functions is referred to as D-finite functions. There are many symbolic algorithms (and implementations thereof) to operate with these objects exactly. Recently, we have extended this notion to a more general class that also allows for good symbolic handling: differentially definable functions. In this paper, we give an overview on what is currently known about this new class.},
volume = {21},
number = {1},
pages = {e202100178--e2021001},
isbn_issn = {?},
year = {2021},
editor = { Gesellschaft für Angewandte Mathematik und Mechanik (GAMM)},
refereed = {yes},
length = {4},
conferencename = {Special Issue: 91st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM)},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.202100178}
}
[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.
[bib]
@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]

Holonomic relations for modular functions and forms: First guess, then prove

Peter Paule, Cristian-Silviu Radu

International Journal of Number Theory 17(3), pp. 713-759. 2021. World Scientific, 1793-0421. [doi]
[bib]
@article{RISC6279,
author = {Peter Paule and Cristian-Silviu Radu},
title = {{Holonomic relations for modular functions and forms: First guess, then prove}},
language = {english},
journal = {International Journal of Number Theory},
volume = {17},
number = {3},
pages = {713--759},
publisher = {World Scientific},
isbn_issn = {1793-0421},
year = {2021},
refereed = {yes},
length = {47},
url = {https://doi.org/10.1142/S1793042120400278}
}
[Paule]

Contiguous Relations and Creative Telescoping

Peter Paule

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Bluemlein and C. Schneider (ed.), Texts and Monographs in Symbolic Computation , pp. -. 2021. Springer, ISBN 978-3-030-80218-9. To appear. [pdf]
[bib]
@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}
}

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