## Members

## Koustav Banerjee

## Abilio de Freitas

## Nikolai Fadeev

## Ralf Hemmecke

## Philipp Nuspl

## Peter Paule: Director

## Veronika Pillwein

## Cristian-Silviu Radu

## Carsten Schneider

## Nicolas Smoot

## Ongoing Projects

### Computer Algebra for Multi-Loop Feynman Integrals

### Partition Congruences by the Localization Method

### SAGEX – Scattering Amplitudes: from Geometry to Experiment

### Extension of Algorithms for D-finite functions [DK15]

### Computer Algebra and Combinatorial Inequalities [FWF SFB F050-07]

### Computer Algebra for Nested Sums and Products [FWF SFB F050-09]

### Partition Analysis [SFB F050-06]

### Computer Algebra Tools for Special Functions [DK6]

## Software

### Asymptotics

#### A Mathematica Package for Computing Asymptotic Series Expansions of Univariate Holonomic Sequences

This package is part of the RISCErgoSum bundle. The Asymptotics package provides a command for computing asymptotic series expansions of solutions of P-finite recurrence equations. ...

### Bibasic Telescope

#### A Mathematica Implementation of a Generalization of Gosper's Algorithm to Bibasic Hypergeometric Summation

This package is part of the RISCErgoSum bundle. pqTelescope is a Mathematica implementation of a generalization of Gosper’s algorithm to indefinite bibasic hypergeometric summation. The package has been developed by Axel Riese, a former member of the RISC Combinatorics group. ...

### Dependencies

#### A Mathematica Package for Computing Algebraic Relations of C-finite Sequences and Multi-Sequences

This package is part of the RISCErgoSum bundle. For any tuple f_1, f_2,..., f_r of sequences, the set of multivariate polynomials p such that p(f1(n),f2(n),...,fr(n))=0 for all points n forms ...

### DiffTools

#### A Mathematica Implementation of several Algorithms for Solving Linear Difference Equations with Polynomial Coefficients

DiffTools is a Mathematica implementation for solving linear difference equations with polynomial coefficients. It contains an algorithm for finding polynomial solutions (by Marko Petkovsek), the algorithm by Sergei Abramov for finding rational solutions, the algorithm of Mark van Hoeij for ...

Paul Kaineder 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. ...

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 ...

A 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. ...

### fastZeil

#### The Paule/Schorn Implementation of Gosper’s and Zeilberger’s Algorithms

This package is part of the RISCErgoSum bundle. With Gosper’s algorithm you can find closed forms for indefinite hypergeometric sums. If you do not succeed, then you may use Zeilberger’s algorithm to come up with a recurrence relation for that ...

### GeneratingFunctions

#### A Mathematica Package for Manipulations of Univariate Holonomic Functions and Sequences

This package is part of the RISCErgoSum bundle. GeneratingFunctions is a Mathematica package for manipulations of univariate holonomic functions and sequences. ...

### GenOmega

#### A Mathematica Implementation of Guo-Niu Han's General Algorithm for MacMahon's Partition Analysis

This package is part of the RISCErgoSum bundle. GenOmega is a Mathematica implementation of Guo-Niu Han’s general Algorithm for MacMahon’s Partition Analysis carried out by Manuela Wiesinger, a master student of the RISC Combinatorics group. Partition Analysis is a computational ...

### Guess

#### A Mathematica Package for Guessing Multivariate Recurrence Equations

This package is part of the RISCErgoSum bundle. The Guess package provides commands for guessing multivariate recurrence equations, as well as for efficiently guessing minimal order univariate recurrence, differential, or algebraic equations given the initial terms of a sequence or ...

### HarmonicSums

#### A Mathematica Package for dealing with Harmonic Sums, Generalized Harmonic Sums and Cyclotomic Sums and their related Integral Representations

The HarmonicSums package by Jakob Ablinger allows to deal with nested sums such as harmonic sums, S-sums, cyclotomic sums and cyclotmic S-sums as well as iterated integrals such as harmonic polylogarithms, multiple polylogarithms and cyclotomic polylogarithms in an algorithmic fashion. ...

### HolonomicFunctions

#### A Mathematica Package for dealing with Multivariate Holonomic Functions, including Closure Properties, Summation, and Integration

This package is part of the RISCErgoSum bundle. The HolonomicFunctions package allows to deal with multivariate holonomic functions and sequences in an algorithmic fashion. For this purpose the package can compute annihilating ideals and execute closure properties (addition, multiplication, substitutions) ...

math4ti2.m is an interface package, allowing the execution of zsolve of the package 4ti2 from within Mathematica notebooks. The package is written by Ralf Hemmecke and Silviu Radu. Licence This program is free software: you can redistribute it and/or ...

### ModularGroup

#### A Mathematica Package providing Basic Algorithms and Visualization Routines related to the Modular Group, e.g. for Drawing the Tessellation of the Upper Half-Plane

ModularGroup.m is a Mathematica package which has been developed in the course of the diploma thesis Computer Algebra and Analysis: Complex Variables Visualized, carried out at the Research Institute for Symbolic Computation (RISC) of the Johannes Kepler University Linz ...

### MultiIntegrate

#### The MultiIntegrate package allows to compute multi-dimensional integrals over hyperexponential integrands in terms of (generalized) harmonic sums.

The MultiIntegrate package allows to compute multi-dimensional integrals over hyperexponential integrands in terms of (generalized) harmonic sums. This package uses variations and extensions of the multivariate Alkmkvist-Zeilberger algorithm. Registration and Legal Notices The source code for this package is password ...

### MultiSum

#### A Mathematica Package for Proving Hypergeometric Multi-Sum Identities

This package is part of the RISCErgoSum bundle. MultiSum is a Mathematica package for proving hypergeometric multi-sum identities. It uses an efficient generalization of Sister Celine’s technique to find a homogeneous polynomial recurrence relation for the sum. The package has ...

Omega is a Mathematica implementation of MacMahon’s Partition Analysis carried out by Axel Riese, a Postdoc of the RISC Combinatorics group. It has been developed together with George E. Andrews and Peter Paule within the frame of a project initiated ...

### ore_algebra

#### A Sage Package for doing Computations with Ore Operators

The ore_algebra package provides an implementation of Ore algebras for Sage. The main features for the most common instances include basic arithmetic and actions; gcrd and lclm; D-finite closure properties; natural transformations between related algebras; guessing; desingularization; solvers for polynomials, ...

### OreSys

#### A Mathematica Implementation of several Algorithms for Uncoupling Systems of Linear Ore Operator Equations

This package is part of the RISCErgoSum bundle. OreSys is a Mathematica package for uncoupling systems of linear Ore operator equations. It offers four algorithms for reducing systems of differential or (q-)difference equations to higher order equations in a single ...

### PermGroup

#### A Mathematica Package for Permutation Groups, Group Actions and Polya Theory

PermGroup is a Mathematica package dealing with permutation groups, group actions and Polya theory. The package has been developed by Thomas Bayer, a former student of the RISC Combinatorics group. ...

### 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 ...

The QEta package is a collection of programs written in the FriCAS computer algebra system that allow to compute with Dedekind eta-functions and related q-series where q=exp(2 π i τ). Furthermore, we provide a number of functions connected to the ...

### qFunctions

#### The qFunctions package is a Mathematica package for q-series and partition theory applications.

The qFunctions package by Jakob Ablinger and Ali K. Uncu is a Mathematica package for q-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for q-shift equations and recurrences ...

### qGeneratingFunctions

#### A Mathematica Package for Manipulations of Univariate q-Holonomic Functions and Sequences

This package is part of the RISCErgoSum bundle. The qGeneratingFunctions package provides commands for manipulating q-holonomic sequences and power series. ...

### qMultiSum

#### A Mathematica Package for Proving q-Hypergeometric Multi-Sum Identities

This package is part of the RISCErgoSum bundle. qMultiSum is a Mathematica package for proving q-hypergeometric multiple summation identities. The package has been developed by Axel Riese, a former member of the RISC Combinatorics group. ...

### qZeil

#### A Mathematica Implementation of q-Analogues of Gosper's and Zeilberger's Algorithm

This package is part of the RISCErgoSum bundle. qZeil is a Mathematica implementation of q-analogues of Gosper’s and Zeilberger’s algorithm for proving and finding indefinite and definite q-hypergeometric summation identities. The package has been developed by Axel Riese, a former ...

RaduRK is a Mathematica implementation of 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 ...

### RatDiff

#### A Mathematica Implementation of Mark van Hoeij's Algorithm for Finding Rational Solutions of Linear Difference Equations

RatDiff is a Mathematica implementation of Mark van Hoeij's algorithm for finding rational solutions of linear difference equations. The package has been developed by Axel Riese, a Postdoc of the RISC Combinatorics group during a stay at the University of ...

### RLangGFun

#### A Maple Implementation of the Inverse Schützenberger Methodology

The inverse Schützenberger methodology transforms a rational generating function into a (pseudo-) regular expression for a corresponding regular language, and is based on Soittola's Theorem about the N-rationality of a formal power series. It is implemented in the Maple package ...

### Sigma

#### A Mathematica Package for Discovering and Proving Multi-Sum Identities

Sigma is a Mathematica package that can handle multi-sums in terms of indefinite nested sums and products. The summation principles of Sigma are: telescoping, creative telescoping and recurrence solving. The underlying machinery of Sigma is based on difference field theory. ...

Singular.m is an interface package, allowing the execution of Singular functions from Mathematica notebooks, written by Manuel Kauers and Viktor Levandovskyy. ...

### Stirling

#### A Mathematica Package for Computing Recurrence Equations of Sums Involving Stirling Numbers or Eulerian Numbers

This package is part of the RISCErgoSum bundle. The Stirling package provides a command for computing recurrence equations of sums involving Stirling numbers or Eulerian numbers. ...

### SumCracker

#### A Mathematica Implementation of several Algorithms for Identities and Inequalities of Special Sequences, including Summation Problems

This package is part of the RISCErgoSum bundle. The SumCracker package contains routines for manipulating a large class of sequences (admissible sequences). It can prove identities and inequalities for these sequences, simplify expressions, evaluate symbolic sums, and solve certain difference ...

### Zeilberger

#### A Maxima Implementation of Gosper's and Zeilberger's Algorithm

Zeilberger is an implementatian for the free and open source Maxima computer algebra system of Gosper's and Zeilberger's algorithm for proving and finding indefinite and definite hypergeometric summation identities. The package has been developed by Fabrizio Caruso, a former Ph. ...

## Publications

### 2022

### 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]**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}

}

### 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]**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)}

}

### 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]**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}

}

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

#### J. Blümlein, C. Schneider

Technical report no. 22-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2022. arXiv:2203.13015 [hep-th]. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6495,

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.},

number = {22-03},

year = {2022},

month = {March},

note = {arXiv:2203.13015 [hep-th]},

keywords = {Feynman integrals, computer algebra, special functions, linear differential equations, linear difference integrals},

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)}

}

### 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

arxiv.2203.13011 [hep-th]. Technical report, 2022. [url]**techreport**{RISC6496,

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},

year = {2022},

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 = {15},

url = {https://arxiv.org/abs/2203.13011}

}

### The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms

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

To appear in Nuclear Physics B(22-01), pp. ?-?. February 2022. ISSN 0550-3213. arXiv:2202.03216 [hep-ph]. [doi]**article**{RISC6497,

author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},

title = {{The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms}},

language = {english},

abstract = {We calculate the unpolarized and polarized two--loop massless off--shell operator matrix elements in QCD to $O(ep)$ in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations upto four--loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin--$N$ space or by harmonic polylogarithms in $z$--space. We also perform comparisons to the literature. },

journal = {To appear in Nuclear Physics B},

number = {22-01},

pages = {?--?},

isbn_issn = {ISSN 0550-3213},

year = {2022},

month = {February},

note = {arXiv:2202.03216 [hep-ph]},

refereed = {yes},

keywords = {QCD, Operator Matrix Element, 2-loop Feynman diagrams, computer algebra, large moment method},

length = {0},

url = {https://doi.org/10.35011/risc.22-01}

}

### 2021

### 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]**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}

}

### 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]**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}

}

### 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]**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}

}

### 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]**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}

}

### 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]**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)}

}

### 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)211, pp. e202100178-e2021001. 2021. [doi]**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}

}

### 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.**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}

}

### 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]**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}

}

### 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]**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}

}

### MacMahon's partition analysis XIII: Schmidt type partitions and modular forms

#### George E. Andrews and Peter Paule

Journal of Number Theory, pp. 95-119. 2021. Elsevier, ISSN 0022-314X. [doi] [pdf]**article**{RISC6373,

author = {George E. Andrews and Peter Paule },

title = {{MacMahon's partition analysis XIII: Schmidt type partitions and modular forms}},

language = {english},

abstract = {In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to $n$ is equal to $p(n)$, the number of partitions of $n$. The object of this paper is to provide a context for this result which leads directly to many other theorems of this nature and which can be viewed as a continuation of our work on elongated partition diamonds. Again generating functions are infinite products built by the Dedekind eta function which, in turn, lead to interesting arithmetic theorems and conjectures for the related partition functions. },

journal = {Journal of Number Theory},

pages = {95--119},

publisher = {Elsevier},

isbn_issn = {ISSN 0022-314X},

year = {2021},

refereed = {yes},

length = {18},

url = {http://doi.org/10.1016/j.jnt.2021.09.008},

type = {open access}

}

### An algorithm to prove holonomic differential equations for modular forms

#### Peter Paule, Cristian-Silviu Radu

In: Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019., Bostan A., Raschel K. (ed.), Springer Proceedings in Mathematics & Statistics 373, pp. 367-420. 2021. Springer, Cham, 978-3-030-84303-8. [doi]**incollection**{RISC6395,

author = {Peter Paule and Cristian-Silviu Radu},

title = {{An algorithm to prove holonomic differential equations for modular forms}},

booktitle = {{Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019.}},

language = {english},

abstract = {Express a modular form $g$ of positive weight locally in terms of a modular function $h$ as$y(h)$, say. Then $y(h)$ as a function in $h$ satisfiesa holonomic differential equation; i.e., one which islinear with coefficients being polynomials in $h$.This fact traces back to Gau{ss} and has beenpopularized prominently by Zagier. Using holonomicprocedures, computationally it is often straightforwardto derive such differential equations as conjectures.In the spirit of the ``first guess, then prove'' paradigm,we present a new algorithm to prove such conjectures.},

series = {Springer Proceedings in Mathematics & Statistics},

volume = {373},

pages = {367--420},

publisher = {Springer, Cham},

isbn_issn = {978-3-030-84303-8},

year = {2021},

editor = {Bostan A. and Raschel K. },

refereed = {yes},

keywords = {modular functions, modular forms, holonomic differential equations, holonomic functions and sequences, Fricke-Klein relations, $q$-series, partition congruences},

sponsor = {FWF SFB F50},

length = {54},

url = {https://doi.org/10.1007/978-3-030-84304-5_16}

}

### Recursion relations for hp-FEM Element Matrices on quadrilaterals

#### Haubold Tim, Pillwein Veronika, Beuchler Sven

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)211, pp. e202100200-e2021002. 2021. [doi]**inproceedings**{RISC6412,

author = {Haubold Tim and Pillwein Veronika and Beuchler Sven},

title = {{Recursion relations for hp-FEM Element Matrices on quadrilaterals}},

booktitle = {{Proceedings in Applied Mathematics and Mechanics (PAMM)}},

language = {english},

abstract = {Abstract In this paper we consider higher order shape functions for finite elements on quadrilaterals. Using tensor products of suitable Jacobi polynomials, it can be proved that the corresponding mass and stiffness matrices are sparse with respect to polynomial degree p . Due to the orthogonal relations between Jacobi polynomials the exact values of the entries of mass and stiffness matrix can be determined. Using symbolic computation, we can find simple recurrence relations which allow us to compute the remaining nonzero entries in optimal arithmetic complexity. Besides the H1 case also the conformal basis functions for H(Div) and H(Curl) are investigated.},

volume = {21},

number = {1},

pages = {e202100200--e2021002},

isbn_issn = {?},

year = {2021},

editor = {Gesellschaft für Angewandte Mathematik und Mechanik (GAMM)},

refereed = {yes},

length = {2},

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.202100200}

}

### Recursion formulas for integrated products of Jacobi polynomials

#### Sven Beuchler, Tim Haubold, Veronika Pillwein

arXiv. Technical report, 2021. [url]**techreport**{RISC6413,

author = {Sven Beuchler and Tim Haubold and Veronika Pillwein},

title = {{Recursion formulas for integrated products of Jacobi polynomials}},

language = {english},

year = {2021},

institution = {arXiv},

length = {25},

url = {https://arxiv.org/abs/2105.08989}

}

### Computing an order complete basis for M∞(N) and applications

#### Mark van Hoeij and Cristian-Silviu Radu

In: Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019, Bostan A., Raschel K. (ed.), Springer Proceedings in Mathematics & Statistics 373, pp. 355-366. 2021. Springer, Cham, 978-3-030-84303-8. [doi]**incollection**{RISC6396,

author = {Mark van Hoeij and Cristian-Silviu Radu },

title = {{Computing an order complete basis for M∞(N) and applications}},

booktitle = {{Transcendence in Algebra, Combinatorics, Geometry and Number Theory. TRANS 2019}},

language = {english},

series = {Springer Proceedings in Mathematics & Statistics},

volume = {373},

pages = {355--366},

publisher = {Springer, Cham},

isbn_issn = {978-3-030-84303-8},

year = {2021},

editor = {Bostan A. and Raschel K.},

refereed = {no},

length = {12},

url = {https://doi.org/10.1007/978-3-030-84304-5_15}

}