## 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

## 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}

}

### 2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family

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

Technical report no. 23-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2023. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6713,

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

title = {{2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family}},

language = {english},

abstract = {Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and present a second congruence family by powers of 7 which we conjecture, and which may be amenable to similar techniques.},

number = {23-10},

year = {2023},

month = {August},

keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},

length = {35},

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)}

}

### Analytic results on the massive three-loop form factors: quarkonic contributions

#### J. Bluemlein, A. De Freitas, P. Marquard, N. Rana, C. Schneider

Physical Review D to appear, pp. ?-?. July 2023. ISSN 2470-0029. arXiv:2307.02983 [hep-ph]. [doi]**article**{RISC6742,

author = {J. Bluemlein and A. De Freitas and P. Marquard and N. Rana and C. Schneider},

title = {{Analytic results on the massive three-loop form factors: quarkonic contributions}},

language = {english},

abstract = {The quarkonic contributions to the three--loop heavy-quark form factors for vector, axial-vector, scalar and pseudoscalar currents are described by closed form difference equations for the expansion coefficients in the limit of small virtualities $q^2/m^2$. A part of the contributions can be solved analytically and expressed in terms of harmonic and cyclotomic harmonic polylogarithms and square-root valued iterated integrals. Other contributions obey equations which are not first--order factorizable. For them still infinite series expansions around the singularities of the form factors can be obtained by matching the expansions at intermediate points and using differential equations which are obeyed directly by the form factors and are derived by guessing algorithms. One may determine all expansion coefficients for $q^2/m^2 rightarrow infty$ analytically in terms of multiple zeta values. By expanding around the threshold and pseudo--threshold, the corresponding constants are multiple zeta values supplemented by a finite amount of new constants, which can be computed at high precision. For a part of these coefficients, the infinite series in front of these constants may be even resummed into harmonic polylogarithms. In this way, one obtains a deeper analytic description of the massive form factors, beyond their pure numerical evaluation. The calculations of these analytic results are based on sophisticated computer algebra techniques. We also compare our results with numerical results in the literature.},

journal = {Physical Review D},

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN 2470-0029},

year = {2023},

month = {July},

note = {arXiv:2307.02983 [hep-ph]},

refereed = {yes},

keywords = {form factor, Feynman diagram, computer algebra, holonomic properties, difference equations, differential equations, symbolic summation, numerical matching, analytic continuation, guessing, PSLQ},

length = {92},

url = {https://doi.org/10.35011/risc.23-08}

}

### Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering

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

In: Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology , Giulio Falcioni (ed.), PoS RADCOR2023046, pp. 1-7. June 2023. ISSN 1824-8039. arXiv:2306.16550 [hep-ph]. [doi]**inproceedings**{RISC6748,

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 = {{Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering}},

booktitle = {{Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology }},

language = {english},

abstract = {We report on recent progress in calculating the three loop QCD corrections of the heavy flavor contributions in deep--inelastic scattering and the massive operator matrix elements of the variable flavor number scheme. Notably we deal with the operator matrix elements $A_{gg,Q}^{(3)}$ and $A_{Qg}^{(3)}$ and technical steps to their calculation. In particular, a new method to obtain the inverse Mellin transform without computing the corresponding $N$--space expressions is discussed.},

series = {PoS},

volume = {RADCOR2023},

number = {046},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2023},

month = {June},

note = {arXiv:2306.16550 [hep-ph]},

editor = {Giulio Falcioni},

refereed = {no},

keywords = {deep-inelastic scattering, 3-loop Feynman diagrams, (inverse) Mellin transform, binomial sums},

length = {7},

url = { https://doi.org/10.22323/1.432.0046 }

}

### 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)}

}

### Algorithms for linear recurrence sequences

#### P. Nuspl

Johannes Kepler University Linz. PhD Thesis. 2023. [pdf]**phdthesis**{RISC6711,

author = {P. Nuspl},

title = {{Algorithms for linear recurrence sequences}},

language = {english},

year = {2023},

translation = {0},

school = {Johannes Kepler University Linz},

length = {129}

}

### 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" in press, 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 = {in press},

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: ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, Gabriela Jeronimo (ed.), pp. 498-507. July 2023. ACM, ISBN 9798400700392. 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 = {{ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

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.},

pages = {498--507},

publisher = {ACM},

isbn_issn = {ISBN 9798400700392},

year = {2023},

month = {July},

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

editor = {Gabriela Jeronimo},

refereed = {yes},

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

length = {10},

url = {https://doi.org/10.1145/3597066.3597073}

}

### Error bounds for the asymptotic expansion of the partition function

#### Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider

Rocky Mt J Math to appear, pp. ?-?. 2023. ISSN: 357596. arXiv:2209.07887 [math.NT]. [doi]**article**{RISC6710,

author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},

title = {{Error bounds for the asymptotic expansion of the partition function}},

language = {english},

abstract = {Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation. },

journal = {Rocky Mt J Math },

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN: 357596},

year = {2023},

note = {arXiv:2209.07887 [math.NT]},

refereed = {yes},

keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},

length = {43},

url = {https://doi.org/10.35011/risc.22-13}

}

### Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package

#### Johannes Bluemlein, Nikolai Fadeev, Carsten Schneider

ACM Communications in Computer Algebra 57(2), pp. 31-34. June 2023. ISSN:1932-2240. arXiv:2308.06042 [hep-ph]. [doi]**article**{RISC6740,

author = {Johannes Bluemlein and Nikolai Fadeev and Carsten Schneider},

title = {{Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package}},

language = {english},

abstract = {Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums. },

journal = {ACM Communications in Computer Algebra},

volume = {57},

number = {2},

pages = {31--34},

isbn_issn = {ISSN:1932-2240},

year = {2023},

month = {June},

note = {arXiv:2308.06042 [hep-ph]},

refereed = {yes},

keywords = {Mellin transform, asymptotic expansions, nested sums, nested integrals, computer algebra},

length = {4},

url = {https://doi.org/10.1145/3614408.3614410}

}

### 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

### Inequalities for the partition function arising from truncated theta series

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

Technical report no. 22-20 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6704,

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

title = {{Inequalities for the partition function arising from truncated theta series}},

language = {english},

abstract = {Positivity questions related to the partition function arising from classical theta identities have been studied in the combinatorial and q-series framework. Two such identities that emerge from truncation of Euler’s pentagonal number theorem and an identity due toGauss are the predominant ones among others. In this paper, we prove the asymptotic growth of coefficients of truncation of theta series directly from inequalities for the shifted partition function.},

number = {22-20},

year = {2022},

month = {August},

keywords = {Partitions, theta series, asymptotics.},

length = {12},

license = {CC BY 4.0 International},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### 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}

}

### Ramanujan's theta functions and parity of parts and cranks of partitions

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

Technical report no. 22-19 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. To appear in Annals of Combinatorics. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6703,

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

title = {{Ramanujan's theta functions and parity of parts and cranks of partitions}},

language = {english},

abstract = {In this paper we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation to the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.},

number = {22-19},

year = {2022},

month = {August},

keywords = {Ramanujan’s theta functions, partitions, parity of parts, cranks.},

length = {13},

type = {To appear in Annals of Combinatorics},

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)}

}