Members
Koustav Banerjee
Abilio de Freitas
Nikolai Fadeev
Yiman Gao
Ralf Hemmecke
Peter Paule: Director 2009-2023
Veronika Pillwein
Cristian-Silviu Radu
Carsten Schneider: Director
Ongoing Projects
Symbolic Summation for Computer Science
Project Lead: Carsten Schneider
Efficient Algorithms for Guessing, Inequalities, Summation
Project Lead: Manuel Kauers
Software
Asymptotics
A Mathematica Package for Computing Asymptotic Series Expansions of Univariate Holonomic Sequences
This package is part of the RISCErgoSum bundle. The Asymptotics package provides a command for computing asymptotic series expansions of solutions of P-finite recurrence equations. ...
Authors: Manuel Kauers
MoreSoftware WebsiteBibasic 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
MoreSoftware WebsiteDependencies
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 ...
Authors: Manuel Kauers, Burkhard Zimmermann
MoreSoftware WebsiteDiffTools
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 ...
Authors: Christian Weixlbaumer
MoreSoftware WebsitePaul 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:
MoreThis 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 ...
Authors: Burkhard Zimmermann
MoreSoftware WebsiteA Mathematica package based on Sigma that tries to evaluate automatically multi-sums to expressions in terms of indefinite nested sums defined over (q-)hypergeometric products. ...
Authors: Carsten Schneider
MoreSoftware WebsitefastZeil
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 ...
Authors: Peter Paule, Markus Schorn
MoreSoftware WebsiteGeneratingFunctions
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. ...
Authors: Christian Mallinger
MoreSoftware WebsiteGenOmega
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 ...
Authors: Manuela Wiesinger-Widi
MoreSoftware WebsiteGuess
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
MoreSoftware WebsiteHarmonicSums
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. ...
Authors: Jakob Ablinger
MoreHolonomicFunctions
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) ...
Authors: Christoph Koutschan
MoreSoftware Websitemath4ti2.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 ...
Authors: Ralf Hemmecke, Cristian-Silviu Radu
MoreSoftware WebsiteModularGroup
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 ...
Authors: Thomas Ponweiser
MoreSoftware WebsiteMultiIntegrate
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 ...
Authors: Jakob Ablinger
MoreMultiSum
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 ...
Authors: Axel Riese, Kurt Wegschaider
MoreSoftware WebsiteOmega 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
MoreSoftware Websiteore_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 ...
Authors: Stefan Gerhold
MoreSoftware WebsitePermGroup
A Mathematica Package for Permutation Groups, Group Actions and Polya Theory
PermGroup is a Mathematica package dealing with permutation groups, group actions and Polya theory. The package has been developed by Thomas Bayer, a former student of the RISC Combinatorics group. ...
Authors: Thomas Bayer
MoreSoftware WebsitePLDESolver
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 ...
Authors: Carsten Schneider, Jakob Ablinger
MorePositiveSequence
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
MoreSoftware WebsiteQEta
A FriCAS package to compute with q-expansions of modular 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
MoreSoftware WebsiteqFunctions
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 ...
Authors: Jakob Ablinger, Ali Uncu
MoreqGeneratingFunctions
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. ...
Authors: Christoph Koutschan
MoreSoftware WebsiteqMultiSum
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
MoreSoftware WebsiteqZeil
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
MoreSoftware WebsiteRaduRK 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
MoreRatDiff
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
MoreSoftware WebsiteRLangGFun
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 ...
Authors: Christoph Koutschan
MoreSoftware WebsiteSigma
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. ...
Authors: Carsten Schneider
MoreSoftware WebsiteSingular.m is an interface package, allowing the execution of Singular functions from Mathematica notebooks, written by Manuel Kauers and Viktor Levandovskyy. ...
Authors: Manuel Kauers, Viktor Levandovskyy
MoreSoftware WebsiteStirling
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
MoreSoftware WebsiteSumCracker
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
MoreSoftware WebsiteZeilberger
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. ...
Authors: Fabrizio Caruso
MoreSoftware WebsitePublications
2026
[de Freitas]
The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$
J. Ablinger, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, Kay Schoenwald
Journal of High Energy Physics 2026(111), pp. 1-52. 2026. ISSN 1029-8479. arXiv:2510.09403 [hep-ph]. [doi]@article{RISC7198,
author = {J. Ablinger and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and Kay Schoenwald},
title = {{The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$}},
language = {english},
abstract = {We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 %$ of the full textcolor{blue}{$O(T_F^2)$} and textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.},
journal = {Journal of High Energy Physics},
volume = {2026},
number = {111},
pages = {1--52},
isbn_issn = {ISSN 1029-8479},
year = {2026},
note = {arXiv:2510.09403 [hep-ph]},
refereed = {yes},
length = {52},
url = {https://doi.org/10.1007/JHEP01(2026)111}
}
author = {J. Ablinger and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and Kay Schoenwald},
title = {{The two-mass contributions to the three-loop massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$}},
language = {english},
abstract = {We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $tilde{A}_{Qg}^{(3)}$ and $Delta tilde{A}_{Qg}^{(3)}$ in $x$-space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to $N = 2000 (3000)$ by an independent method, to which we compare the results in $x$-space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about $50 %$ of the full textcolor{blue}{$O(T_F^2)$} and textcolor{blue}{$O(T_F^3)$} terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.},
journal = {Journal of High Energy Physics},
volume = {2026},
number = {111},
pages = {1--52},
isbn_issn = {ISSN 1029-8479},
year = {2026},
note = {arXiv:2510.09403 [hep-ph]},
refereed = {yes},
length = {52},
url = {https://doi.org/10.1007/JHEP01(2026)111}
}
[de Freitas]
The single-mass variable flavor number scheme at three-loop order
J. Ablinger, A. Behring, J. Bluemlein, d, A. De Freitas, A. von Manteuffel, C. Schneider, and K. Schoenwald
Journal of High Energy Physics 2026(248), pp. 0-33. 2026. SSN 1029-8479. arXiv:2510.02175 [hep-ph]. [doi]@article{RISC7229,
author = {J. Ablinger and A. Behring and J. Bluemlein and d and A. De Freitas and A. von Manteuffel and C. Schneider and and K. Schoenwald},
title = {{The single-mass variable flavor number scheme at three-loop order}},
language = {english},
abstract = {The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $overline{rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided. },
journal = {Journal of High Energy Physics},
volume = {2026},
number = {248},
pages = {0--33},
isbn_issn = {SSN 1029-8479},
year = {2026},
note = {arXiv:2510.02175 [hep-ph]},
refereed = {yes},
length = {34},
url = {https://doi.org/10.1007/JHEP03(2026)248}
}
author = {J. Ablinger and A. Behring and J. Bluemlein and d and A. De Freitas and A. von Manteuffel and C. Schneider and and K. Schoenwald},
title = {{The single-mass variable flavor number scheme at three-loop order}},
language = {english},
abstract = {The matching relations in the unpolarized and polarized variable flavor number scheme at three-loop order are presented in the single-mass case. They describe the process of massive quarks becoming light at large virtualities $Q^2$. In this framework, heavy-quark parton distributions can be defined. Numerical results are presented on the matching relations in the case of the single-mass variable flavor number scheme for the light parton, charm and bottom quark distributions. These relations are process independent. In the polarized case we generally work in the Larin scheme. To two-loop order we present the polarized massive OMEs also in the $overline{rm MS}$ scheme. Fast numerical codes for the single-mass massive operator matrix elements are provided. },
journal = {Journal of High Energy Physics},
volume = {2026},
number = {248},
pages = {0--33},
isbn_issn = {SSN 1029-8479},
year = {2026},
note = {arXiv:2510.02175 [hep-ph]},
refereed = {yes},
length = {34},
url = {https://doi.org/10.1007/JHEP03(2026)248}
}
[de Freitas]
The heavy quark-antiquark asymmetry in the variable flavor number scheme
A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald
Physics Letters B 876(140411), pp. 1-8. 2026. ISSN 1873-2445. arXiv:2512.13508 [hep-ph]. [doi]@article{RISC7238,
author = {A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The heavy quark-antiquark asymmetry in the variable flavor number scheme}},
language = {english},
abstract = {The twist-2 heavy-quark and antiquark distributions, as defined in the variable flavor number scheme, turn out to be different due to QCD corrections from three-loop onward. This is caused by terms containing the color factor $d_{abc} d^{abc}$ in the heavy-flavor massive pure-singlet operator matrix elements (OMEs) $A^{rm PS, s, (3)}_{Qq}$ for odd moments in the unpolarized case and for $Delta A^{rm PS, s, (3)}_{Qq}$ for even moments in the polarized case. The dependence on the factorization scale of the OMEs is ruled by the anomalous dimensions $gamma^{rm NS, s, (2)}_{qq}$ and $Delta gamma^{rm NS, s, (2)}_{qq}$. The polarized calculations are performed in the Larin scheme. We compute the corresponding three-loop heavy-flavor distributions $(Delta) f_Q(x,Q^2) - (Delta) f_{overline{Q}}(x,Q^2)$. Compared to the sum of the heavy-quark and antiquark parton distributions, their difference is small, however, non-vanishing. },
journal = {Physics Letters B},
volume = {876},
number = {140411},
pages = {1--8},
isbn_issn = {ISSN 1873-2445},
year = {2026},
note = {arXiv:2512.13508 [hep-ph]},
refereed = {yes},
length = {8},
url = {https://doi.org/10.1016/j.physletb.2026.140411}
}
author = {A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The heavy quark-antiquark asymmetry in the variable flavor number scheme}},
language = {english},
abstract = {The twist-2 heavy-quark and antiquark distributions, as defined in the variable flavor number scheme, turn out to be different due to QCD corrections from three-loop onward. This is caused by terms containing the color factor $d_{abc} d^{abc}$ in the heavy-flavor massive pure-singlet operator matrix elements (OMEs) $A^{rm PS, s, (3)}_{Qq}$ for odd moments in the unpolarized case and for $Delta A^{rm PS, s, (3)}_{Qq}$ for even moments in the polarized case. The dependence on the factorization scale of the OMEs is ruled by the anomalous dimensions $gamma^{rm NS, s, (2)}_{qq}$ and $Delta gamma^{rm NS, s, (2)}_{qq}$. The polarized calculations are performed in the Larin scheme. We compute the corresponding three-loop heavy-flavor distributions $(Delta) f_Q(x,Q^2) - (Delta) f_{overline{Q}}(x,Q^2)$. Compared to the sum of the heavy-quark and antiquark parton distributions, their difference is small, however, non-vanishing. },
journal = {Physics Letters B},
volume = {876},
number = {140411},
pages = {1--8},
isbn_issn = {ISSN 1873-2445},
year = {2026},
note = {arXiv:2512.13508 [hep-ph]},
refereed = {yes},
length = {8},
url = {https://doi.org/10.1016/j.physletb.2026.140411}
}
[de Freitas]
The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x, Q^2)$ and $g_1(x, Q^2)$
J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schönwald
Physics Letters B 878(140540), pp. 1-8. 2026. ISSN 0370-2693. arXiv:2509.16124 [hep-ph]. [doi]@article{RISC7241,
author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schönwald},
title = {{The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$}},
language = {english},
journal = {Physics Letters B},
volume = {878},
number = {140540},
pages = {1--8},
isbn_issn = {ISSN 0370-2693},
year = {2026},
note = {arXiv:2509.16124 [hep-ph]},
refereed = {yes},
length = {8},
url = {https://doi.org/10.1016/j.physletb.2026.140540}
}
author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schönwald},
title = {{The three-loop single-mass heavy-flavor corrections to the structure functions $F_2(x,Q^2)$ and $g_1(x,Q^2)$}},
language = {english},
journal = {Physics Letters B},
volume = {878},
number = {140540},
pages = {1--8},
isbn_issn = {ISSN 0370-2693},
year = {2026},
note = {arXiv:2509.16124 [hep-ph]},
refereed = {yes},
length = {8},
url = {https://doi.org/10.1016/j.physletb.2026.140540}
}
[de Freitas]
The variable flavor number scheme to three-loop order
J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald
Technical report no. 26-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2026. DESY 26-064, RISC Report number 26-06, CERN-TH-2026-113, MPP-2026-89, PoS (LL2026) 025. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC7247,
author = {J. Ablinger and A. Behring and J.~Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The variable flavor number scheme to three-loop order}},
language = {english},
abstract = {We describe the variable flavor number scheme to three-loop order, which modifies the massless parton densities by single- and two-mass effects and introduces heavy-quark parton distribution functions for charm and bottom. A renormalization group analysis shows the validity of this picture at large scales $Q^2$, where it resembles the non-power-suppressed heavy-flavor corrections completely. We also provide numerical implementations of a series of charged and neutral current Wilson coefficients.},
number = {26-06},
year = {2026},
month = {July},
note = {DESY 26--064, RISC Report number 26-06, CERN-TH-2026-113, MPP-2026-89, PoS (LL2026) 025},
keywords = {variable flavor number scheme, heavy-quark parton distribution function, computer algebra, numerical implementation},
length = {11},
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)}
}
author = {J. Ablinger and A. Behring and J.~Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The variable flavor number scheme to three-loop order}},
language = {english},
abstract = {We describe the variable flavor number scheme to three-loop order, which modifies the massless parton densities by single- and two-mass effects and introduces heavy-quark parton distribution functions for charm and bottom. A renormalization group analysis shows the validity of this picture at large scales $Q^2$, where it resembles the non-power-suppressed heavy-flavor corrections completely. We also provide numerical implementations of a series of charged and neutral current Wilson coefficients.},
number = {26-06},
year = {2026},
month = {July},
note = {DESY 26--064, RISC Report number 26-06, CERN-TH-2026-113, MPP-2026-89, PoS (LL2026) 025},
keywords = {variable flavor number scheme, heavy-quark parton distribution function, computer algebra, numerical implementation},
length = {11},
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)}
}
[de Freitas]
The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients
J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald
In: 17th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology (RADCOR2025), M. C. Kumar, Narayan Rana, Vajravelu Ravindran, Satyajit Seth, Ambresh Shivaji (ed.), POS 497076, pp. 1-16. 2026. ISSN 1824-8039 . arXiv:2602.10334 [hep-ph]. [doi]@inproceedings{RISC7258,
author = {J. Ablinger and A. Behring and J.~Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients}},
booktitle = {{ 17th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology (RADCOR2025)}},
language = {english},
abstract = {We report on the three-loop unpolarized and polarized massive operator matrix elements, with single- and two-mass corrections, and the associated deep-inelastic massive Wilson coefficients in the region $Q^2 gg m_Q^2$, the calculation of which has been completed recently. We also provide fast and precise numerical representations ofthe massless Wilson coefficients, splitting functions to tree-loop order, and target-mass corrections in $x$-space well suited for QCD-fitting codes.},
series = {POS},
volume = {497},
number = {076},
pages = {1--16},
isbn_issn = {ISSN 1824-8039 },
year = {2026},
note = {arXiv:2602.10334 [hep-ph]},
editor = {M. C. Kumar and Narayan Rana and Vajravelu Ravindran and Satyajit Seth and Ambresh Shivaji},
refereed = {no},
keywords = { three-loop unpolarized and polarized massive operator matrix elements, deep-inelastic scattering, computer algebra, special functions},
length = {16},
url = {https://doi.org/10.35011/risc.26-01}
}
author = {J. Ablinger and A. Behring and J.~Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The complete three-loop unpolarized and polarized massive operator matrix elements and asymptotic Wilson coefficients}},
booktitle = {{ 17th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology (RADCOR2025)}},
language = {english},
abstract = {We report on the three-loop unpolarized and polarized massive operator matrix elements, with single- and two-mass corrections, and the associated deep-inelastic massive Wilson coefficients in the region $Q^2 gg m_Q^2$, the calculation of which has been completed recently. We also provide fast and precise numerical representations ofthe massless Wilson coefficients, splitting functions to tree-loop order, and target-mass corrections in $x$-space well suited for QCD-fitting codes.},
series = {POS},
volume = {497},
number = {076},
pages = {1--16},
isbn_issn = {ISSN 1824-8039 },
year = {2026},
note = {arXiv:2602.10334 [hep-ph]},
editor = {M. C. Kumar and Narayan Rana and Vajravelu Ravindran and Satyajit Seth and Ambresh Shivaji},
refereed = {no},
keywords = { three-loop unpolarized and polarized massive operator matrix elements, deep-inelastic scattering, computer algebra, special functions},
length = {16},
url = {https://doi.org/10.35011/risc.26-01}
}
[Raab]
Universal truth of operator statements via ideal membership
Georg Regensburger, Clemens Hofstadler, Clemens Raab
Journal of Pure and Applied Algebra 230(108221), pp. 0-0. 2026. 0022-4049. [doi]@article{RISC7252,
author = {Georg Regensburger and Clemens Hofstadler and Clemens Raab},
title = {{Universal truth of operator statements via ideal membership}},
language = {english},
abstract = {We introduce a framework for proving statements about linear operators by verification of ideal membership in a free algebra. More specifically, arbitrary first-order statements about identities of morphisms in preadditive semicategories can be treated. We present a semi-decision procedure for validity of such formulas based on computations with noncommutative polynomials. These algebraic computations automatically incorporate linearity and benefit from efficient ideal membership procedures.In the framework, domains and codomains of operators are modelled using many-sorted first-order logic. To eliminate quantifiers and function symbols from logical formulas, we apply Herbrand's theorem and Ackermann's reduction. The validity of the resulting formulas is shown to be equivalent to finitely many ideal memberships of noncommutative polynomials. We explain all relevant concepts and discuss computational aspects. Furthermore, we illustrate our framework by proving concrete operator statements assisted by our computer algebra software.},
journal = {Journal of Pure and Applied Algebra},
volume = {230},
number = {108221},
pages = {0--0},
isbn_issn = {0022-4049},
year = {2026},
refereed = {yes},
length = {40},
url = {https://doi.org/10.1016/j.jpaa.2026.108221}
}
author = {Georg Regensburger and Clemens Hofstadler and Clemens Raab},
title = {{Universal truth of operator statements via ideal membership}},
language = {english},
abstract = {We introduce a framework for proving statements about linear operators by verification of ideal membership in a free algebra. More specifically, arbitrary first-order statements about identities of morphisms in preadditive semicategories can be treated. We present a semi-decision procedure for validity of such formulas based on computations with noncommutative polynomials. These algebraic computations automatically incorporate linearity and benefit from efficient ideal membership procedures.In the framework, domains and codomains of operators are modelled using many-sorted first-order logic. To eliminate quantifiers and function symbols from logical formulas, we apply Herbrand's theorem and Ackermann's reduction. The validity of the resulting formulas is shown to be equivalent to finitely many ideal memberships of noncommutative polynomials. We explain all relevant concepts and discuss computational aspects. Furthermore, we illustrate our framework by proving concrete operator statements assisted by our computer algebra software.},
journal = {Journal of Pure and Applied Algebra},
volume = {230},
number = {108221},
pages = {0--0},
isbn_issn = {0022-4049},
year = {2026},
refereed = {yes},
length = {40},
url = {https://doi.org/10.1016/j.jpaa.2026.108221}
}
[Regensburger]
Refuting noncommutative ideal memberschip via matrix certificates
Georg Regensburger, Clemens Hofstadler, Peter Krug
In: Proceedings of ISSAC 2026, Christoph Koutschan, Alin Bostan, Clement pernet, Thi Xuan Vu (ed.), pp. 209-218. 2026. 979-8-4007-2595-1. [doi]@inproceedings{RISC7251,
author = {Georg Regensburger and Clemens Hofstadler and Peter Krug},
title = {{Refuting noncommutative ideal memberschip via matrix certificates}},
booktitle = {{Proceedings of ISSAC 2026}},
language = {english},
abstract = {The ideal membership problem in free algebras is undecidable in general. More precisely, while membership can always be verified in finite time (e.g., via noncommutative Gröbner bases), non-membership is undecidable in general.In this work, we introduce matrix certificates for refuting ideal membership of (commutative and) noncommutative polynomials. Such certificates are matrix evaluations that vanish on the generators of an ideal but not on a given candidate polynomial. For commutative polynomials, a perfect Nullstellensatz guarantees the existence of such certificates with commuting square matrices. For noncommutative polynomials, certificates may require non-square matrices or may not exist at all. To handle evaluations on non-square matrices, we use quivers and their matrix representations.We have implemented an approach for finding matrix certificates by ansatz in SageMath and demonstrate its effectiveness on different examples. Our method relies on the ability to efficiently find one (simple) solution to a system of commutative polynomial equations, which we do by combining SAT solving with Hensel lifting. Our experiments suggest that, in practice, ideal (non-)membership can be efficiently decided and certified.},
pages = {209--218},
isbn_issn = {979-8-4007-2595-1},
year = {2026},
editor = {Christoph Koutschan and Alin Bostan and Clement pernet and Thi Xuan Vu},
refereed = {yes},
length = {10},
url = {https://dl.acm.org/doi/10.1145/3815436.3815447}
}
author = {Georg Regensburger and Clemens Hofstadler and Peter Krug},
title = {{Refuting noncommutative ideal memberschip via matrix certificates}},
booktitle = {{Proceedings of ISSAC 2026}},
language = {english},
abstract = {The ideal membership problem in free algebras is undecidable in general. More precisely, while membership can always be verified in finite time (e.g., via noncommutative Gröbner bases), non-membership is undecidable in general.In this work, we introduce matrix certificates for refuting ideal membership of (commutative and) noncommutative polynomials. Such certificates are matrix evaluations that vanish on the generators of an ideal but not on a given candidate polynomial. For commutative polynomials, a perfect Nullstellensatz guarantees the existence of such certificates with commuting square matrices. For noncommutative polynomials, certificates may require non-square matrices or may not exist at all. To handle evaluations on non-square matrices, we use quivers and their matrix representations.We have implemented an approach for finding matrix certificates by ansatz in SageMath and demonstrate its effectiveness on different examples. Our method relies on the ability to efficiently find one (simple) solution to a system of commutative polynomial equations, which we do by combining SAT solving with Hensel lifting. Our experiments suggest that, in practice, ideal (non-)membership can be efficiently decided and certified.},
pages = {209--218},
isbn_issn = {979-8-4007-2595-1},
year = {2026},
editor = {Christoph Koutschan and Alin Bostan and Clement pernet and Thi Xuan Vu},
refereed = {yes},
length = {10},
url = {https://dl.acm.org/doi/10.1145/3815436.3815447}
}
[Regensburger]
Parametrized systems of generalized polynomial inequalities via linear algebra and convex geometry
Georg Regensburger, Stefan Müller
Positivity 30(4), pp. 0-0. 2026. 1385-1292. [url]@article{RISC7253,
author = {Georg Regensburger and Stefan Müller},
title = {{Parametrized systems of generalized polynomial inequalities via linear algebra and convex geometry}},
language = {english},
abstract = {We provide fundamental results on positive solutions to parametrized systems of generalized polynomial inequalities (with real exponents and positive parameters), including generalized polynomial equations. In doing so, we also offer a new perspective on fewnomials and (generalized) mass-action systems. We find that geometric objects, rather than matrices, determine generalized polynomial systems: a bounded set/“polytope” P (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency d, is crucial. As our main result, we rewrite polynomial inequalities in terms of d binomial equations on P, involving d monomials in the parameters. In particular, we establish an explicit bijection between the original solution set and the solution set on P via exponentiation. (i) Our results apply to any generalized polynomial system. (ii) The dependency d and the dimension of P indicate the complexity of a system. (iii) Our results are based on methods from linear algebra and convex/polyhedral geometry, and the solution set on P can be further studied using methods from analysis such as sign-characteristic functions (introduced in this work). We illustrate our results (in particular, the relevant geometric objects) through three examples from real fewnomial and reaction network theory. For two mass-action systems, we parametrize the set of equilibria and the region for multistationarity, respectively, and even for univariate trinomials, we offer new insights: We provide a “solution formula” involving discriminants and “roots”.},
journal = {Positivity},
volume = {30},
number = {4},
pages = {0--0},
isbn_issn = {1385-1292},
year = {2026},
refereed = {yes},
length = {26},
url = {https://link.springer.com/article/10.1007/s11117-025-01158-4}
}
author = {Georg Regensburger and Stefan Müller},
title = {{Parametrized systems of generalized polynomial inequalities via linear algebra and convex geometry}},
language = {english},
abstract = {We provide fundamental results on positive solutions to parametrized systems of generalized polynomial inequalities (with real exponents and positive parameters), including generalized polynomial equations. In doing so, we also offer a new perspective on fewnomials and (generalized) mass-action systems. We find that geometric objects, rather than matrices, determine generalized polynomial systems: a bounded set/“polytope” P (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency d, is crucial. As our main result, we rewrite polynomial inequalities in terms of d binomial equations on P, involving d monomials in the parameters. In particular, we establish an explicit bijection between the original solution set and the solution set on P via exponentiation. (i) Our results apply to any generalized polynomial system. (ii) The dependency d and the dimension of P indicate the complexity of a system. (iii) Our results are based on methods from linear algebra and convex/polyhedral geometry, and the solution set on P can be further studied using methods from analysis such as sign-characteristic functions (introduced in this work). We illustrate our results (in particular, the relevant geometric objects) through three examples from real fewnomial and reaction network theory. For two mass-action systems, we parametrize the set of equilibria and the region for multistationarity, respectively, and even for univariate trinomials, we offer new insights: We provide a “solution formula” involving discriminants and “roots”.},
journal = {Positivity},
volume = {30},
number = {4},
pages = {0--0},
isbn_issn = {1385-1292},
year = {2026},
refereed = {yes},
length = {26},
url = {https://link.springer.com/article/10.1007/s11117-025-01158-4}
}
[Schneider]
The $q$-extension of iterated integrals and nested sums in quantum field theory
J. Bluemlein, A.M. Gavrilik, O. Mykhailiv, C. Schneider
Technical report no. 26-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2026. arXiv:2608.02702[math-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC7257,
author = {J. Bluemlein and A.M. Gavrilik and O. Mykhailiv and C. Schneider},
title = {{The $q$-extension of iterated integrals and nested sums in quantum field theory}},
language = {english},
abstract = {Analytic calculations of zero- and single-scale quantities in perturbative quantum fieldtheory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincar'e iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums,nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sumscontaining central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involvedalphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions arequite different form the corresponding $mu$-extended functions.},
number = {26-11},
year = {2026},
month = {August},
note = {arXiv:2608.02702[math-ph]},
keywords = {q-difference equations, q-differential eqations, q-iterative integrals, q-iterative sums, holonomic closure properties, recurrence solving, quantum field theory},
length = {40},
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)}
}
author = {J. Bluemlein and A.M. Gavrilik and O. Mykhailiv and C. Schneider},
title = {{The $q$-extension of iterated integrals and nested sums in quantum field theory}},
language = {english},
abstract = {Analytic calculations of zero- and single-scale quantities in perturbative quantum fieldtheory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincar'e iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums,nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sumscontaining central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involvedalphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions arequite different form the corresponding $mu$-extended functions.},
number = {26-11},
year = {2026},
month = {August},
note = {arXiv:2608.02702[math-ph]},
keywords = {q-difference equations, q-differential eqations, q-iterative integrals, q-iterative sums, holonomic closure properties, recurrence solving, quantum field theory},
length = {40},
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]
A Survey on Symbolic Summation in Difference Rings
C. Schneider
In: Computer Algebra in Scientific Computing, Boulier, F., Mou, C., Sadykov, T.M., Uncu, A.K. (ed.), Computer Algebra in Scientific Computing. CASC 2026 16844, pp. 1-32. 2026. Springer Nature Switzerland, ISBN 978-3-032-34585-1. RISC Report Series 26-07, https://doi.org/10.35011/risc.26-07. [doi]@inproceedings{RISC7259,
author = {C. Schneider},
title = {{A Survey on Symbolic Summation in Difference Rings}},
booktitle = {{Computer Algebra in Scientific Computing}},
language = {english},
abstract = {This survey article provides an overview of the fundamental principles used to simplify multi-sums into indefinite nested sums over hypergeometric products in the setting of difference rings. We place special emphasis on the algorithmic translation between hypergeometric sums and the formal difference ring setting. Furthermore, we detail the core summation paradigms of telescoping, creative telescoping, and recurrence solving within difference rings, illustrating these techniques and their underlying algorithms with concrete examples.},
series = { Computer Algebra in Scientific Computing. CASC 2026},
volume = {16844},
pages = {1--32},
publisher = {Springer Nature Switzerland},
isbn_issn = {ISBN 978-3-032-34585-1},
year = {2026},
note = {RISC Report Series 26-07, https://doi.org/10.35011/risc.26-07},
editor = {Boulier and F. and Mou and C. and Sadykov and T.M. and Uncu and A.K.},
refereed = {yes},
keywords = {Difference ring, telescoping, creative telescoping, parameterized telescoping, recurrence solving.},
length = {32},
url = {https://doi.org/10.1007/978-3-032-34586-8_1}
}
author = {C. Schneider},
title = {{A Survey on Symbolic Summation in Difference Rings}},
booktitle = {{Computer Algebra in Scientific Computing}},
language = {english},
abstract = {This survey article provides an overview of the fundamental principles used to simplify multi-sums into indefinite nested sums over hypergeometric products in the setting of difference rings. We place special emphasis on the algorithmic translation between hypergeometric sums and the formal difference ring setting. Furthermore, we detail the core summation paradigms of telescoping, creative telescoping, and recurrence solving within difference rings, illustrating these techniques and their underlying algorithms with concrete examples.},
series = { Computer Algebra in Scientific Computing. CASC 2026},
volume = {16844},
pages = {1--32},
publisher = {Springer Nature Switzerland},
isbn_issn = {ISBN 978-3-032-34585-1},
year = {2026},
note = {RISC Report Series 26-07, https://doi.org/10.35011/risc.26-07},
editor = {Boulier and F. and Mou and C. and Sadykov and T.M. and Uncu and A.K.},
refereed = {yes},
keywords = {Difference ring, telescoping, creative telescoping, parameterized telescoping, recurrence solving.},
length = {32},
url = {https://doi.org/10.1007/978-3-032-34586-8_1}
}
2025
[Chen]
A Unified Reduction for Hypergeometric and $q$-Hypergeometric Creative Telescoping
Shaoshi Chen, Hao Du, Yiman Gao, Hui Huang, Ziming Li
The Ramanujan J. 68(14), pp. 1-39. 2025. ISSN 1572-9303. arXiv:2501.03837 [cs.SC]. [doi] [pdf]@article{RISC7154,
author = {Shaoshi Chen and Hao Du and Yiman Gao and Hui Huang and Ziming Li},
title = {{A Unified Reduction for Hypergeometric and $q$-Hypergeometric Creative Telescoping}},
language = {english},
journal = {The Ramanujan J.},
volume = {68},
number = {14},
pages = {1--39},
isbn_issn = {ISSN 1572-9303},
year = {2025},
note = {arXiv:2501.03837 [cs.SC]},
refereed = {yes},
length = {39},
url = {https://doi.org/10.1007/s11139-025-01164-w}
}
author = {Shaoshi Chen and Hao Du and Yiman Gao and Hui Huang and Ziming Li},
title = {{A Unified Reduction for Hypergeometric and $q$-Hypergeometric Creative Telescoping}},
language = {english},
journal = {The Ramanujan J.},
volume = {68},
number = {14},
pages = {1--39},
isbn_issn = {ISSN 1572-9303},
year = {2025},
note = {arXiv:2501.03837 [cs.SC]},
refereed = {yes},
length = {39},
url = {https://doi.org/10.1007/s11139-025-01164-w}
}
[Fadeev]
Computer algebra for special functions
Nikolai Fadeev
RISC, Johannes Kepler University Linz. PhD Thesis. May 2025.@phdthesis{RISC7201,
author = {Nikolai Fadeev},
title = {{Computer algebra for special functions}},
language = {english},
abstract = {Calculations done in different mathematical areas — such as computer algebra,combinatorics, number theory, differential equations — and physical areas — suchas particle physics — give rise to a plethora of problems involving special functionsthat need to be dealt with efficiently. In this PhD, we concentrated on two suchparticular problems.In the first part of this PhD thesis, we explored the relation between iteratedbinomial sums, an extension of general harmonic sums, and their integral representations, in order to compute their asymptotic expansions. To do that in a fullyautomatic way, we created a dedicated package, RICA. Using Mellin representations,we first formalised and extended a classical recursive method to compute Mellininverses of such sums, and together with it implemented several methods to compute asymptotic expansions of such integrals. In the process, we introduced andexplored a new class of functions related to Mellin convolutions. Those allowed usto automatically compute asymptotic expansions for more general classes of sumsin a new and efficient way, while providing a way to get symbolic representationsfor the constants appearing in the calculation of the Mellin inversions.In the second part of this PhD thesis, we studied first order inhomogeneous systems of differential equations involving an extra parameter epsilon coming from particlephysics computations. Since usually those systems could only be solved up to someorder in epsilon, we aimed at developing a method to optimise the solving task of suchsystems. We studied an approach centered on the minimisation of the epsilon-order in theexpansion of the inhomogeneous part. In particular, we proposed a method basedon separating the system in smaller subsystems called triangularization, beforeanalysing each of those individually using dffierent uncoupling schemes, selectedpriorization of equations and through comparisons of epsilon-orders. This method hasbeen implemented in a package called SystemAnalysis.},
year = {2025},
month = {May},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {292}
}
author = {Nikolai Fadeev},
title = {{Computer algebra for special functions}},
language = {english},
abstract = {Calculations done in different mathematical areas — such as computer algebra,combinatorics, number theory, differential equations — and physical areas — suchas particle physics — give rise to a plethora of problems involving special functionsthat need to be dealt with efficiently. In this PhD, we concentrated on two suchparticular problems.In the first part of this PhD thesis, we explored the relation between iteratedbinomial sums, an extension of general harmonic sums, and their integral representations, in order to compute their asymptotic expansions. To do that in a fullyautomatic way, we created a dedicated package, RICA. Using Mellin representations,we first formalised and extended a classical recursive method to compute Mellininverses of such sums, and together with it implemented several methods to compute asymptotic expansions of such integrals. In the process, we introduced andexplored a new class of functions related to Mellin convolutions. Those allowed usto automatically compute asymptotic expansions for more general classes of sumsin a new and efficient way, while providing a way to get symbolic representationsfor the constants appearing in the calculation of the Mellin inversions.In the second part of this PhD thesis, we studied first order inhomogeneous systems of differential equations involving an extra parameter epsilon coming from particlephysics computations. Since usually those systems could only be solved up to someorder in epsilon, we aimed at developing a method to optimise the solving task of suchsystems. We studied an approach centered on the minimisation of the epsilon-order in theexpansion of the inhomogeneous part. In particular, we proposed a method basedon separating the system in smaller subsystems called triangularization, beforeanalysing each of those individually using dffierent uncoupling schemes, selectedpriorization of equations and through comparisons of epsilon-orders. This method hasbeen implemented in a package called SystemAnalysis.},
year = {2025},
month = {May},
translation = {0},
school = {RISC, Johannes Kepler University Linz},
length = {292}
}
[Gao]
Complete Reduction for Derivatives in a Primitive Tower
Hao Du, Yiman Gao, Wenqiao Li and Ziming Li
In: Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation (ISSAC’25, Santiago Laplagne (ed.), pp. 42-51. 2025. 979-8-4007-2075-8/25/07.@inproceedings{RISC7191,
author = {Hao Du and Yiman Gao and Wenqiao Li and Ziming Li},
title = {{Complete Reduction for Derivatives in a Primitive Tower}},
booktitle = {{ Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation (ISSAC’25}},
language = {english},
pages = {42--51},
isbn_issn = {979-8-4007-2075-8/25/07},
year = {2025},
editor = {Santiago Laplagne},
refereed = {yes},
length = {10}
}
author = {Hao Du and Yiman Gao and Wenqiao Li and Ziming Li},
title = {{Complete Reduction for Derivatives in a Primitive Tower}},
booktitle = {{ Proceedings of the 2025 International Symposium on Symbolic and Algebraic Computation (ISSAC’25}},
language = {english},
pages = {42--51},
isbn_issn = {979-8-4007-2075-8/25/07},
year = {2025},
editor = {Santiago Laplagne},
refereed = {yes},
length = {10}
}
[Hemmecke]
Computer-assisted construction of Ramanujan-Sato series for 1 over pi
Ralf Hemmecke, Peter Paule, Cristian-Silviu Radu
Technical report no. 25-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2025. Licensed under CC BY 4.0 International. [doi] [pdf] [pdf]@techreport{RISC7134,
author = {Ralf Hemmecke and Peter Paule and Cristian-Silviu Radu},
title = {{Computer-assisted construction of Ramanujan-Sato series for 1 over pi}},
language = {english},
abstract = {Referring to ideasof Takeshi Sato, Yifan Yang in~cite{YangDE} described a construction ofseries for $1$ over $pi$ startingwith a pair $(g,h)$, where $g$ is a modular formof weight $2$ and $h$ is a modular function; i.e.,a modular form of weight zero. In this article we present an algorithmicversion,called ``Sato construction''. Series for $1/pi$ obtained this way will becalled ``Ramanujan-Sato''series. Famous series fit into this definition, for instance, Ramanujan'sseries used by Gosperand the series used by the Chudnovsky brothersfor computing millions of digits of $pi$. Weshow that these series are induced by membersof infinite families of Sato triples $(N, gamma_N,tau_N)$ where $N>1$ is an integer and $gamma_N$ a $2times 2$ matrixsatisfying $gamma_N tau_N=N tau_N$ for$tau_N$ being an element from the upper half of thecomplex plane.In addition to procedures for guessingand proving from the holonomic toolbox togetherwiththe algorithm ``ModFormDE'', as describedin~cite{PPSR:ModFormDE1}, a central roleis played by the algorithm ``MultiSamba'',an extension ofSamba (``subalgebra module basis algorithm'') originating fromcite{Radu_RamanujanKolberg_2015} and cite{Hemmecke}.With thehelp of MultiSamba one canfind and prove evaluations of modular functions,at imaginary quadratic points, in terms of nested algebraic expressions.As a consequence,all the series for $1/pi$ constructed withthe help of MultiSamba are proven completelyin a rigorous non-numerical manner.},
number = {25-01},
year = {2025},
month = {January},
keywords = {modular forms and functions, holonomic differential equations, Ramanujan-Sato series for 1 over pi, MultiSamba algorithm},
length = {58},
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)}
}
author = {Ralf Hemmecke and Peter Paule and Cristian-Silviu Radu},
title = {{Computer-assisted construction of Ramanujan-Sato series for 1 over pi}},
language = {english},
abstract = {Referring to ideasof Takeshi Sato, Yifan Yang in~cite{YangDE} described a construction ofseries for $1$ over $pi$ startingwith a pair $(g,h)$, where $g$ is a modular formof weight $2$ and $h$ is a modular function; i.e.,a modular form of weight zero. In this article we present an algorithmicversion,called ``Sato construction''. Series for $1/pi$ obtained this way will becalled ``Ramanujan-Sato''series. Famous series fit into this definition, for instance, Ramanujan'sseries used by Gosperand the series used by the Chudnovsky brothersfor computing millions of digits of $pi$. Weshow that these series are induced by membersof infinite families of Sato triples $(N, gamma_N,tau_N)$ where $N>1$ is an integer and $gamma_N$ a $2times 2$ matrixsatisfying $gamma_N tau_N=N tau_N$ for$tau_N$ being an element from the upper half of thecomplex plane.In addition to procedures for guessingand proving from the holonomic toolbox togetherwiththe algorithm ``ModFormDE'', as describedin~cite{PPSR:ModFormDE1}, a central roleis played by the algorithm ``MultiSamba'',an extension ofSamba (``subalgebra module basis algorithm'') originating fromcite{Radu_RamanujanKolberg_2015} and cite{Hemmecke}.With thehelp of MultiSamba one canfind and prove evaluations of modular functions,at imaginary quadratic points, in terms of nested algebraic expressions.As a consequence,all the series for $1/pi$ constructed withthe help of MultiSamba are proven completelyin a rigorous non-numerical manner.},
number = {25-01},
year = {2025},
month = {January},
keywords = {modular forms and functions, holonomic differential equations, Ramanujan-Sato series for 1 over pi, MultiSamba algorithm},
length = {58},
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)}
}
[Hemmecke]
An Algorithm to Compute Algebraic Relations Between Modular Functions
Ralf Hemmecke, Peter Paule, Cristian-Silviu Radu
Technical report no. 25-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). November 2025. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC7197,
author = {Ralf Hemmecke and Peter Paule and Cristian-Silviu Radu},
title = {{An Algorithm to Compute Algebraic Relations Between Modular Functions}},
language = {english},
abstract = {The existence of an algebraic relation between two modular functions,in short: a modular equation, is implied by a classical fact from thetheory of compact Riemann surfaces. In this article, we present a new,purely algebraic proof of the existence of modular equations. Oursetting consists of an algorithmic framework which is based on areduction procedure for tuples of formal Laurent series. The resultingalgorithm MultiSamba (“sub-algebra module basis algorithm”) is part ofHemmecke's computer algebra package QEta which has been implemented inFriCAS, a general purpose computer algebra system which is freelyavailable as open source. QEta is a powerful tool-box for actualcomputations. For example, MultiSamba has been used forcomputer-assisted discovery and proofs of Ramanujan-Sato series. Inthis article, we describe the mathematics underlying the MultiSambaalgorithm. Moreover, we explain in detail how MultiSamba works for thederivationof a well-known modular equation betweenthe modular $\lambda$-function and the Klein $j$function.Other examples of the automatic discovery and proving of modularequations include identities by Alladi and others, which suggestrelations of Ramanujan-G\"ollnitz-Gordon type as another promisingarea of MultiSamba application.},
number = {25-09},
year = {2025},
month = {November},
keywords = {modular functions, multisamba, modular equations},
length = {23},
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)}
}
author = {Ralf Hemmecke and Peter Paule and Cristian-Silviu Radu},
title = {{An Algorithm to Compute Algebraic Relations Between Modular Functions}},
language = {english},
abstract = {The existence of an algebraic relation between two modular functions,in short: a modular equation, is implied by a classical fact from thetheory of compact Riemann surfaces. In this article, we present a new,purely algebraic proof of the existence of modular equations. Oursetting consists of an algorithmic framework which is based on areduction procedure for tuples of formal Laurent series. The resultingalgorithm MultiSamba (“sub-algebra module basis algorithm”) is part ofHemmecke's computer algebra package QEta which has been implemented inFriCAS, a general purpose computer algebra system which is freelyavailable as open source. QEta is a powerful tool-box for actualcomputations. For example, MultiSamba has been used forcomputer-assisted discovery and proofs of Ramanujan-Sato series. Inthis article, we describe the mathematics underlying the MultiSambaalgorithm. Moreover, we explain in detail how MultiSamba works for thederivationof a well-known modular equation betweenthe modular $\lambda$-function and the Klein $j$function.Other examples of the automatic discovery and proving of modularequations include identities by Alladi and others, which suggestrelations of Ramanujan-G\"ollnitz-Gordon type as another promisingarea of MultiSamba application.},
number = {25-09},
year = {2025},
month = {November},
keywords = {modular functions, multisamba, modular equations},
length = {23},
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]
Creative Telescoping for Hypergeometric Double Sums
P. Paule, C. Schneider
J. Symb. Comput. 128(102394), pp. 1-30. 2025. ISSN: 0747-7171. Symbolic Computation and Combinatorics: A special issue in memory and honor of Marko Petkovšek, edited by Shaoshi Chen, Sergei Abramov, Manuel Kauers, Eugene Zima. [doi]@article{RISC7068,
author = {P. Paule and C. Schneider},
title = {{Creative Telescoping for Hypergeometric Double Sums}},
language = {english},
abstract = {We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.},
journal = {J. Symb. Comput.},
volume = {128},
number = {102394},
pages = {1--30},
isbn_issn = {ISSN: 0747-7171},
year = {2025},
note = {Symbolic Computation and Combinatorics: A special issue in memory and honor of Marko Petkovšek, edited by Shaoshi Chen, Sergei Abramov, Manuel Kauers, Eugene Zima},
refereed = {yes},
keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},
length = {30},
url = {https://doi.org/10.1016/j.jsc.2024.102394}
}
author = {P. Paule and C. Schneider},
title = {{Creative Telescoping for Hypergeometric Double Sums}},
language = {english},
abstract = {We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.},
journal = {J. Symb. Comput.},
volume = {128},
number = {102394},
pages = {1--30},
isbn_issn = {ISSN: 0747-7171},
year = {2025},
note = {Symbolic Computation and Combinatorics: A special issue in memory and honor of Marko Petkovšek, edited by Shaoshi Chen, Sergei Abramov, Manuel Kauers, Eugene Zima},
refereed = {yes},
keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},
length = {30},
url = {https://doi.org/10.1016/j.jsc.2024.102394}
}
[Schneider]
Asymptotics for the reciprocal and shifted quotient of the partition function
Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider
Research in Number Theory 11(101), pp. 1-46. 2025. ISSN 2363-9555. arXiv:2412.02257 [math.NT]. [doi]@article{RISC7184,
author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},
title = {{Asymptotics for the reciprocal and shifted quotient of the partition function}},
language = {english},
abstract = {Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $kin mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest.},
journal = {Research in Number Theory},
volume = {11},
number = {101},
pages = {1--46},
isbn_issn = {ISSN 2363-9555},
year = {2025},
note = { arXiv:2412.02257 [math.NT]},
refereed = {yes},
length = {46},
url = {https://doi.org/10.1007/s40993-025-00678-y}
}
author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},
title = {{Asymptotics for the reciprocal and shifted quotient of the partition function}},
language = {english},
abstract = {Let $p(n)$ denote the partition function. In this paper our main goal is to derive an asymptotic expansion up to order $N$ (for any fixed positive integer $N$) along with estimates for error bounds for the shifted quotient of the partition function, namely $p(n+k)/p(n)$ with $kin mathbb{N}$, which generalizes a result of Gomez, Males, and Rolen. In order to do so, we derive asymptotic expansions with error bounds for the shifted version $p(n+k)$ and the multiplicative inverse $1/p(n)$, which is of independent interest.},
journal = {Research in Number Theory},
volume = {11},
number = {101},
pages = {1--46},
isbn_issn = {ISSN 2363-9555},
year = {2025},
note = { arXiv:2412.02257 [math.NT]},
refereed = {yes},
length = {46},
url = {https://doi.org/10.1007/s40993-025-00678-y}
}
[Schneider]
Telescoping Algorithms for $Sigma^*$-Extensions via Complete Reductions
S. Chen and Y. Gao and H. Huang and C. Schneider
In: Recent Trends in Computer Algebra, Bruno Salvy, Alin Bostan, Mohab Safey El Din, Gilles Villard (ed.), Texts & Monographs in Symbolic Computation , pp. ?-?. 2025. Springer Nature, arXiv:2506.08767 [cs.SC]. [doi]@incollection{RISC7199,
author = {S. Chen and Y. Gao and H. Huang and C. Schneider},
title = {{Telescoping Algorithms for $Sigma^*$-Extensions via Complete Reductions}},
booktitle = {{Recent Trends in Computer Algebra}},
language = {english},
abstract = {A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such thatthe given element is summable if and only if the remainder is equal to zero.In this paper, we present a complete reduction in a tower of $Sigma^*$-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such $Sigma^*$-extensions cover iterative sums such as the harmonic numbers and generalized versions that arise, e.g., in combinatorics, computer science or particle physics. Moreover, we illustrate how these new ideas can be used to reduce the depth of the given sum and provide structural theorems that connect complete reductions to Karr's Fundamental Theorem of symbolic summation.},
series = {Texts & Monographs in Symbolic Computation},
pages = {?--?},
publisher = {Springer Nature},
isbn_issn = {?},
year = {2025},
note = {arXiv:2506.08767 [cs.SC]},
editor = {Bruno Salvy and Alin Bostan and Mohab Safey El Din and Gilles Villard},
refereed = {yes},
length = {35},
url = {https://doi.org/10.35011/risc.25-05}
}
author = {S. Chen and Y. Gao and H. Huang and C. Schneider},
title = {{Telescoping Algorithms for $Sigma^*$-Extensions via Complete Reductions}},
booktitle = {{Recent Trends in Computer Algebra}},
language = {english},
abstract = {A complete reduction on a difference field is a linear operator that enables one to decompose an element of the field as the sum of a summable part and a remainder such thatthe given element is summable if and only if the remainder is equal to zero.In this paper, we present a complete reduction in a tower of $Sigma^*$-extensions that turns to a new efficient framework for the parameterized telescoping problem. Special instances of such $Sigma^*$-extensions cover iterative sums such as the harmonic numbers and generalized versions that arise, e.g., in combinatorics, computer science or particle physics. Moreover, we illustrate how these new ideas can be used to reduce the depth of the given sum and provide structural theorems that connect complete reductions to Karr's Fundamental Theorem of symbolic summation.},
series = {Texts & Monographs in Symbolic Computation},
pages = {?--?},
publisher = {Springer Nature},
isbn_issn = {?},
year = {2025},
note = {arXiv:2506.08767 [cs.SC]},
editor = {Bruno Salvy and Alin Bostan and Mohab Safey El Din and Gilles Villard},
refereed = {yes},
length = {35},
url = {https://doi.org/10.35011/risc.25-05}
}
2024
[de Freitas]
The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$
J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. von Manteuffel, C. Schneider, K. Schoenwald
Nuclear Physics B 999(116427), pp. 1-42. 2024. ISSN 0550-3213. arXiv:2311.00644 [hep-ph]. [doi]@article{RISC6755,
author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The first--order factorizable contributions to the three--loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},
language = {english},
abstract = {The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$contain first--order factorizable and non--first--order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first--order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color--$zeta$ factors for the cases in which also non--first--order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$--space, and correspondingly, on Kummer--Poincar'e and square--root valued alphabets in Bjorken--$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$--space analytically and we also discuss the limitations in the present case to analytically continue the given $N$--space expressions to $N in mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first--order factorizable results over a 17--letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$--space, also containing up to weight {sf w = 5} special constants, which can be rationalized to Kummer--Poincar'e iterated integrals at special arguments. The analytic $x$--space representation requires separate analyses for the intervals $x in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first--order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high--precision numerical solution of the problems.},
journal = {Nuclear Physics B},
volume = {999},
number = {116427},
pages = {1--42},
isbn_issn = {ISSN 0550-3213},
year = {2024},
note = {arXiv:2311.00644 [hep-ph]},
refereed = {yes},
keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, nested integrals, nested sums},
length = {42},
url = {https://doi.org/10.1016/j.nuclphysb.2023.116427}
}
author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The first--order factorizable contributions to the three--loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},
language = {english},
abstract = {The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$contain first--order factorizable and non--first--order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first--order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color--$zeta$ factors for the cases in which also non--first--order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$--space, and correspondingly, on Kummer--Poincar'e and square--root valued alphabets in Bjorken--$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$--space analytically and we also discuss the limitations in the present case to analytically continue the given $N$--space expressions to $N in mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first--order factorizable results over a 17--letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$--space, also containing up to weight {sf w = 5} special constants, which can be rationalized to Kummer--Poincar'e iterated integrals at special arguments. The analytic $x$--space representation requires separate analyses for the intervals $x in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first--order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high--precision numerical solution of the problems.},
journal = {Nuclear Physics B},
volume = {999},
number = {116427},
pages = {1--42},
isbn_issn = {ISSN 0550-3213},
year = {2024},
note = {arXiv:2311.00644 [hep-ph]},
refereed = {yes},
keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, nested integrals, nested sums},
length = {42},
url = {https://doi.org/10.1016/j.nuclphysb.2023.116427}
}