## Members

## Koustav Banerjee

## Abilio de Freitas

## Nikolai Fadeev

## Ralf Hemmecke

## Philipp Nuspl

## Peter Paule: Director

## Veronika Pillwein

## Cristian-Silviu Radu

## Carsten Schneider

## Nicolas Smoot

## Lukas Woegerer

## Ongoing Projects

### Computer Algebra for Multi-Loop Feynman Integrals

### Partition Congruences by the Localization Method

## 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 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. ...

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 ...

### PositiveSequence

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

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

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

### 2024

### Representation of hypergeometric products of higher nesting depths in difference rings

#### E.D. Ocansey, C. Schneider

J. Symb. Comput. 120, pp. 1-50. 2024. ISSN: 0747-7171. arXiv:2011.08775 [cs.SC]. [doi]**article**{RISC6688,

author = {E.D. Ocansey and C. Schneider},

title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},

language = {english},

journal = {J. Symb. Comput.},

volume = {120},

pages = {1--50},

isbn_issn = {ISSN: 0747-7171},

year = {2024},

note = {arXiv:2011.08775 [cs.SC]},

refereed = {yes},

length = {50},

url = {https://doi.org/10.1016/j.jsc.2023.03.002}

}

### 2023

### Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach

#### Koustav Banerjee

Enumerative Combinatorics and Applications 3, pp. 1-4. 2023. ISSN 2710-2335. [doi]**article**{RISC6701,

author = {Koustav Banerjee},

title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},

language = {english},

journal = {Enumerative Combinatorics and Applications},

volume = {3},

pages = {1--4},

isbn_issn = {ISSN 2710-2335},

year = {2023},

refereed = {yes},

length = {5},

url = {https://doi.org/10.54550/ECA2023V3S2R12}

}

### Inequalities for the modified Bessel function of first kind of non-negative order

#### K. Banerjee

Journal of Mathematical Analysis and Applications 524, pp. 1-28. 2023. Elsevier, ISSN 1096-0813. [doi]**article**{RISC6700,

author = {K. Banerjee},

title = {{Inequalities for the modified Bessel function of first kind of non-negative order}},

language = {english},

journal = {Journal of Mathematical Analysis and Applications},

volume = {524},

pages = {1--28},

publisher = {Elsevier},

isbn_issn = {ISSN 1096-0813},

year = {2023},

refereed = {yes},

length = {28},

url = {https://doi.org/10.1016/j.jmaa.2023.127082}

}

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

#### A. Jimenez-Pastor, P. Nuspl, V. Pillwein

Journal of Symbolic Computation 116, pp. 400-424. 2023. ISSN: 0747-7171.**article**{RISC6636,

author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},

title = {{An extension of holonomic sequences: $C^2$-finite sequences}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {116},

pages = {400--424},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

refereed = {yes},

length = {25}

}

### Order bounds for $C^2$-finite sequences

#### M. Kauers, P. Nuspl, V. Pillwein

Technical report no. 23-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6683,

author = {M. Kauers and P. Nuspl and V. Pillwein},

title = {{Order bounds for $C^2$-finite sequences}},

language = {english},

abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},

number = {23-03},

year = {2023},

month = {February},

keywords = {Difference equations, holonomic sequences, closure properties, 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)}

}

### Ramanujan and Computer Algebra

#### Peter Paule

In: Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, K. Alladi, G.E. Andrews, B. Berndt, F. Garvan, K. Ono, P. Paule, S. Ole Warnaar, Ae Ja Yee (ed.), pp. -. 2023. Springer, ISBN x. [pdf]**incollection**{RISC6678,

author = {Peter Paule},

title = {{Ramanujan and Computer Algebra}},

booktitle = {{Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence}},

language = {english},

pages = {--},

publisher = {Springer},

isbn_issn = {ISBN x},

year = {2023},

editor = {K. Alladi and G.E. Andrews and B. Berndt and F. Garvan and K. Ono and P. Paule and S. Ole Warnaar and Ae Ja Yee },

refereed = {yes},

length = {0}

}

### Interview with Peter Paule

#### Toufik Mansour and Peter Paule

Enumerative Combinatorics and Applications ECA 3:1(#S3I1), pp. -. 2023. ISSN 2710-2335. [doi]**article**{RISC6679,

author = {Toufik Mansour and Peter Paule},

title = {{Interview with Peter Paule}},

language = {english},

journal = {Enumerative Combinatorics and Applications },

volume = {ECA 3:1},

number = {#S3I1},

pages = {--},

isbn_issn = {ISSN 2710-2335},

year = {2023},

refereed = {yes},

length = {0},

url = {http://doi.org/10.54550/ECA2023V3S1I1}

}

### Hypergeometric Structures in Feynman Integrals

#### J. Blümlein, C. Schneider, M. Saragnese

Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science" to appear, pp. ?-?. 2023. ISSN 1573-7470. arXiv:2111.15501 [math-ph]. [doi]**article**{RISC6643,

author = {J. Blümlein and C. Schneider and M. Saragnese},

title = {{Hypergeometric Structures in Feynman Integrals}},

language = {english},

abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},

journal = {Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science"},

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN 1573-7470},

year = {2023},

note = {arXiv:2111.15501 [math-ph]},

refereed = {yes},

keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},

length = {55},

url = {https://doi.org/10.1007/s10472-023-09831-8}

}

### Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators

#### C. Schneider

In: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation (Proc. ISSAC 23), Gabriela Jeronimo (ed.), to appear , pp. ?-?. February 2023. arXiv:2302.03563 [cs.SC]. [doi]**inproceedings**{RISC6699,

author = {C. Schneider},

title = {{Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators}},

booktitle = {{Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation (Proc. ISSAC 23)}},

language = {english},

abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $RPiSigma$-ring extensions that are built over general $PiSigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},

series = {to appear},

pages = {?--?},

isbn_issn = {?},

year = {2023},

month = {February},

note = {arXiv:2302.03563 [cs.SC]},

editor = {Gabriela Jeronimo},

refereed = {yes},

keywords = {telescoping, difference rings, reduced denominators, nested sums},

length = {9},

url = {https://doi.org/10.35011/risc.23-01}

}

### A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method

#### N. Smoot

Journal of Number Theory 242, pp. 112-153. January 2023. ISSN 1096-1658. [doi]**article**{RISC6661,

author = {N. Smoot},

title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},

language = {english},

abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},

journal = {Journal of Number Theory},

volume = {242},

pages = {112--153},

isbn_issn = {ISSN 1096-1658},

year = {2023},

month = {January},

refereed = {yes},

length = {42},

url = {https://doi.org/10.1016/j.jnt.2022.07.014}

}

### 2022

### Hook type tableaux and partition identities

#### K. Banerjee, M. G. Dastidar

Notes on Number Theory and Discrete Mathematics 28(4), pp. 635-647. 2022. ISSN 2367–8275. [doi]**article**{RISC6702,

author = {K. Banerjee and M. G. Dastidar},

title = {{Hook type tableaux and partition identities}},

language = {english},

journal = { Notes on Number Theory and Discrete Mathematics },

volume = {28},

number = {4},

pages = {635--647},

isbn_issn = {ISSN 2367–8275},

year = {2022},

refereed = {yes},

length = {13},

url = {https://doi.org/10.7546/nntdm.2022.28.4.635-647}

}

### Hook Type enumeration and parity of parts in partitions

#### K. Banerjee, M. G. Dastidar

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6596, 2022. [pdf]**techreport**{RISC6596,

author = {K. Banerjee and M. G. Dastidar},

title = {{Hook Type enumeration and parity of parts in partitions}},

language = {english},

abstract = {This paper is devoted to study an association between hook type enumeration and counting integer partitions subject to parity of its parts. We shall primarily focus on a result of Andrews in two possible direction. First, we confirm a conjecture of Rubey and secondly, we extend the theorem of Andrews in a more general set up. },

number = {RISC6596},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {8}

}

### Parity biases in partitions and restricted partitions

#### Banerjee Koustav, Bhattacharjee Sreerupa, Dastidar Manosij Ghosh, Mahanta Pankaj Jyoti, Saikia Manjil P

European Journal of Combinatorics 103, pp. 103522-103522. 2022. Elsevier, ISSN 0195-6698. [pdf]**article**{RISC6606,

author = {Banerjee Koustav and Bhattacharjee Sreerupa and Dastidar Manosij Ghosh and Mahanta Pankaj Jyoti and Saikia Manjil P},

title = {{Parity biases in partitions and restricted partitions}},

language = {english},

journal = {European Journal of Combinatorics},

volume = {103},

pages = {103522--103522},

publisher = {Elsevier},

isbn_issn = {ISSN 0195-6698},

year = {2022},

refereed = {yes},

length = {19}

}

### New inequalities for p(n) and log p(n)

#### K. Banerjee, P. Paule, C. S. Radu, W. H. Zeng

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6607, 2022. To appear in the Ramanujan Journal. [pdf]**techreport**{RISC6607,

author = {K. Banerjee and P. Paule and C. S. Radu and W. H. Zeng},

title = {{New inequalities for p(n) and log p(n)}},

language = {english},

number = {RISC6607},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {37},

type = {To appear in the Ramanujan Journal}

}

### An unified framework to prove multiplicative inequalities for the partition function

#### K. Banerjee

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6614, 2022. [pdf]**techreport**{RISC6614,

author = {K. Banerjee},

title = {{An unified framework to prove multiplicative inequalities for the partition function}},

language = {english},

number = {RISC6614},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {24}

}

### The localization method applied to k-elognated plane partitions and divisibily by 5

#### K. Banerjee, N. A. Smoot

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6611, 2022. [pdf]**techreport**{RISC6611,

author = {K. Banerjee and N. A. Smoot},

title = {{The localization method applied to k-elognated plane partitions and divisibily by 5}},

language = {english},

number = {RISC6611},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {40}

}

### Invariants of the quartic binary form and proofs of Chen's conjectures on partition function inequalities

#### K. Banerjee

Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6615, 2022. [pdf]**techreport**{RISC6615,

author = {K. Banerjee},

title = {{Invariants of the quartic binary form and proofs of Chen's conjectures on partition function inequalities}},

language = {english},

number = {RISC6615},

year = {2022},

institution = {Research Institute for Symbolic Computation, JKU, Linz},

length = {31}

}

### The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. Goedicke, A. von Manteuffel, C. Schneider, K. Schoenwald

Journal of High Energy Physics 2022(12, Article 134), pp. 1-55. 2022. ISSN 1029-8479. arXiv:2211.05462 [hep-ph]. [doi]**article**{RISC6632,

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},

language = {english},

abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$--space and iterated integrals over square root--valued alphabets in momentum fraction $x$--space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$--space and derive numerical results. The present results complete the gluon transition matrix elements both of the single-- and double--mass variable flavor number scheme to three--loop order.},

journal = {Journal of High Energy Physics},

volume = {2022},

number = {12, Article 134},

pages = {1--55},

isbn_issn = { ISSN 1029-8479},

year = {2022},

note = {arXiv:2211.05462 [hep-ph]},

refereed = {yes},

keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},

length = {48},

url = {https://doi.org/10.1007/JHEP12(2022)134}

}

### 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)}

}