## Members

## Jakob Ablinger

## Koustav Banerjee

## Nikolai Fadeev

## Ankush Goswami

## Ralf Hemmecke

## Antonio Jimenez Pastor

## Philipp Nuspl

## Evans Doe Ocansey

## Peter Paule

## Veronika Pillwein

## Cristian-Silviu Radu

## Carsten Schneider

## Nicolas Smoot

## Ali Uncu

## Ongoing Projects

### 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 ...

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

### 2021

### Extensions of the AZ-algorithm and the Package MultiIntegrate

#### J. Ablinger

Technical report no. 21-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2021. [pdf]**techreport**{RISC6272,

author = {J. Ablinger},

title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},

language = {english},

number = {21-02},

year = {2021},

length = {25},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 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}

}

### The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra

#### C. Schneider

Mathematics in Computer Science , appeared electronically, pp. ?-?. 2021. ISSN 1661-8289. arXiv:2003.01921 [math.CO]. [url]**article**{RISC6127,

author = {C. Schneider},

title = {{The Absent-Minded Passengers Problem: A Motivating Challenge Solved by Computer Algebra}},

language = {english},

journal = {Mathematics in Computer Science , appeared electronically},

pages = {?--?},

isbn_issn = {ISSN 1661-8289},

year = {2021},

note = {arXiv:2003.01921 [math.CO]},

refereed = {yes},

length = {12},

url = {https://doi.org/10.1007/s11786-020-00494-w}

}

### Three loop heavy quark form factors and their asymptotic behavior

#### J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider

In: To appear in Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018, , pp. ?-?. 2021. arXiv:1906.05829 [hep-ph]. [url]**inproceedings**{RISC6024,

author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Three loop heavy quark form factors and their asymptotic behavior}},

booktitle = {{To appear in Proc.of 23rd DAE-BRNS High Energy Physics Symposium 2018}},

language = {english},

pages = {?--?},

isbn_issn = {?},

year = {2021},

note = {arXiv:1906.05829 [hep-ph]},

editor = {?},

refereed = {yes},

length = {8},

url = {https://arxiv.org/abs/1906.05829}

}

### A case study for ζ(4)

#### Carsten Schneider, Wadim Zudilin

In: Proceedings of the conference 'Transient Transcendence in Transylvania', Alin Bostan and Kilian Raschel (ed.), Proceedings in Mathematics & Statistics , pp. ?-?. 2021. Springer, arXiv:2004.08158 [math.NT]. [url]**incollection**{RISC6210,

author = {Carsten Schneider and Wadim Zudilin},

title = {{A case study for ζ(4)}},

booktitle = {{Proceedings of the conference 'Transient Transcendence in Transylvania'}},

language = {english},

series = {Proceedings in Mathematics & Statistics},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2021},

note = {arXiv:2004.08158 [math.NT]},

editor = {Alin Bostan and Kilian Raschel},

refereed = {no},

length = {0},

url = {https://arxiv.org/abs/2004.08158}

}

### On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ∗-field extensions

#### Sergei A. Abramov, Manuel Bronstein, Marko Petkovšek, Carsten Schneider

J. Symb. Comput. 107, pp. 23-66. 2021. ISSN 0747-7171. arXiv:2005.04944 [cs.SC]. [url]**article**{RISC6224,

author = {Sergei A. Abramov and Manuel Bronstein and Marko Petkovšek and Carsten Schneider},

title = {{On Rational and Hypergeometric Solutions of Linear Ordinary Difference Equations in ΠΣ∗-field extensions}},

language = {english},

journal = {J. Symb. Comput.},

volume = {107},

pages = {23--66},

isbn_issn = {ISSN 0747-7171},

year = {2021},

note = {arXiv:2005.04944 [cs.SC]},

refereed = {yes},

length = {44},

url = {https://doi.org/10.1016/j.jsc.2021.01.002}

}

### The Polarized Transition Matrix Element $A_{g, q}(N)$ of the Variable Flavor Number Scheme at $O(alpha_s^3)$

#### A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, K. Schönwald, and C. Schneider

Nuclear Physics B 964, pp. 115331-115356. 2021. ISSN 0550-3213. arXiv:2101.05733 [hep-ph]. [url]**article**{RISC6278,

author = {A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and K. Schönwald and and C. Schneider},

title = {{The Polarized Transition Matrix Element $A_{g,q}(N)$ of the Variable Flavor Number Scheme at $O(alpha_s^3)$}},

language = {english},

journal = {Nuclear Physics B},

volume = {964},

pages = {115331--115356},

isbn_issn = {ISSN 0550-3213},

year = {2021},

note = {arXiv:2101.05733 [hep-ph]},

refereed = {yes},

length = {26},

url = {https://doi.org/10.1016/j.nuclphysb.2021.115331}

}

### Solving linear difference equations with coefficients in rings with idempotent representations

#### J. Ablinger, C. Schneider

Technical report no. 21-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2021. [pdf]**techreport**{RISC6284,

author = {J. Ablinger and C. Schneider},

title = {{Solving linear difference equations with coefficients in rings with idempotent representations}},

language = {english},

number = {21-04},

year = {2021},

length = {8},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

#### C. Schneider

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Blümlein and C. Schneider (ed.), Texts and Monographs in Symbolic Computuation to appear, pp. ?-?. 2021. Springer, arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03. [url]**incollection**{RISC6287,

author = {C. Schneider},

title = {{Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation}},

booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation},

volume = {to appear},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2021},

note = {arXiv:2102.01471 [cs.SC], RISC-Linz Report Series No. 21-03},

editor = {J. Blümlein and C. Schneider},

refereed = {yes},

length = {55},

url = {https://arxiv.org/abs/2102.01471}

}

### The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering

#### J. Blümlein, A. De Freitas, M. Saragnese, K. Schönwald, C. Schneider

Technical report no. 21-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2021. [pdf]**techreport**{RISC6292,

author = {J. Blümlein and A. De Freitas and M. Saragnese and K. Schönwald and C. Schneider},

title = {{The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering}},

language = {english},

number = {21-06},

year = {2021},

length = {86},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2020

### Proving Two Conjectural Series for $\zeta(7)$ and Discovering More Series for $\zeta(7)$

#### J. Ablinger

In: Mathematical Aspects of Computer and Information Science, D. Slamanig, E. Tsigaridas, Z. Zafeirakopoulos (ed.), pp. 42-47. 2020. Springer International Publishing, 978-3-030-43120-4. [url]**inproceedings**{RISC6102,

author = {J. Ablinger},

title = {{Proving Two Conjectural Series for $\zeta(7)$ and Discovering More Series for $\zeta(7)$}},

booktitle = {{Mathematical Aspects of Computer and Information Science}},

language = {english},

pages = {42--47},

publisher = {Springer International Publishing},

isbn_issn = {978-3-030-43120-4},

year = {2020},

editor = {D. Slamanig and E. Tsigaridas and Z. Zafeirakopoulos},

refereed = {yes},

length = {6},

url = {https://arxiv.org/abs/1908.06631v1}

}

### Subleading logarithmic QED initial state corrections to $e^+e^−\to γ^⁎/Z^{0⁎}$ to $O(\alpha^6L^5)$

#### J. Ablinger, J. Blümlein, A. De Freitas, K. Schönwald

Nuclear Physics B 955, pp. 115045-115045. 2020. ISSN 0550-3213. [url]**article**{RISC6111,

author = {J. Ablinger and J. Blümlein and A. De Freitas and K. Schönwald},

title = {{Subleading logarithmic QED initial state corrections to $e^+e^−\to γ^⁎/Z^{0⁎}$ to $O(\alpha^6L^5)$}},

language = {english},

journal = {Nuclear Physics B},

volume = {955},

pages = {115045--115045},

isbn_issn = { ISSN 0550-3213},

year = {2020},

refereed = {yes},

length = {0},

url = {http://www.sciencedirect.com/science/article/pii/S0550321320301310}

}

### The Sage Package Comb_walks for Walks in the Quarter Plane

#### Antonio Jiménez-Pastor, Alin Bostan, Frédéric Chyzak, Pierre Lairez

ACM Commun. Comput. Algebra 54(2), pp. 30-38. sep 2020. Association for Computing Machinery, New York, NY, USA, 1932-2240. [url]**article**{RISC6282,

author = {Antonio Jiménez-Pastor and Alin Bostan and Frédéric Chyzak and Pierre Lairez},

title = {{The Sage Package Comb_walks for Walks in the Quarter Plane}},

language = {english},

abstract = {We present in this extended abstract a new software designed to work with generating functions that count walks in the quarter plane. With this software we offer a cohesive package that brings together all the required procedures for manipulating these generating functions, as well as a unified interface to deal with them. We also display results that this package offers on a public webpage.},

journal = {ACM Commun. Comput. Algebra},

volume = {54},

number = {2},

pages = {30--38},

publisher = {Association for Computing Machinery},

address = {New York, NY, USA},

isbn_issn = {1932-2240},

year = {2020},

month = {sep},

refereed = {yes},

keywords = {Sage, D-algebraic functions, generating functions, elliptic functions, lattice walks},

length = {9},

url = {https://doi.org/10.1145/3427218.3427220}

}

### Hook Type Tableaux and Partition Identities

#### Koustav Banerjee, Manosij Ghosh Dastidar

Research Institute for Symbolic Computation. Technical report, 2020. Preprint. [pdf]**techreport**{RISC6118,

author = {Koustav Banerjee and Manosij Ghosh Dastidar},

title = {{Hook Type Tableaux and Partition Identities}},

language = {english},

abstract = {In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of the Stanley’s and Elder’s theorem without the use of partition statistics in general. We explain how the principle can be used to prove another interesting theorem on partitions with parts separated by parity, a special case of which is George Andrews’s result in [2].},

year = {2020},

note = {Preprint},

institution = {Research Institute for Symbolic Computation},

length = {19}

}

### First-order factorizable systems of differential equations in one variable

#### N. Fadeev

Technical report no. 20-20 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2020. [pdf]**techreport**{RISC6222,

author = {N. Fadeev},

title = {{First-order factorizable systems of differential equations in one variable}},

language = {english},

number = {20-20},

year = {2020},

length = {29},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### On sums of coefficients of polynomials related to the Borwein conjectures

#### Ankush Goswami, Venkata Raghu Tej Pantangi

Technical report no. 20-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. Ramanujan J. (to appear), May 2020. [pdf]**techreport**{RISC6113,

author = {Ankush Goswami and Venkata Raghu Tej Pantangi},

title = {{On sums of coefficients of polynomials related to the Borwein conjectures}},

language = {english},

abstract = {Recently, Li (Int. J. Number Theory 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci. China. Math. 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $p>3$. In this work, we extend Li's method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $p=3, 5$, the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J. 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $p=2, 3, 5, 7, 11, 13$ and for $p>15$, we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large $n$.},

number = {20-07},

year = {2020},

month = {May},

howpublished = {Ramanujan J. (to appear)},

length = {13},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Some formulae for coefficients in restricted $q$-products

#### Ankush Goswami, Venkata Raghu Tej Pantangi

Technical report no. 20-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. May 2020. [pdf]**techreport**{RISC6114,

author = {Ankush Goswami and Venkata Raghu Tej Pantangi},

title = {{Some formulae for coefficients in restricted $q$-products}},

language = {english},

abstract = {In this paper, we derive some formulae involving coefficients of polynomials which occur quite naturally in the study of restricted partitions. Our method involves a recently discovered sieve technique by Li and Wan (Sci. China. Math. 2010). Based on this method, by considering cyclic groups of different orders we obtain some new results for these coefficients. The general result (see Theorem \ref{main00}) holds for any group of the form $\mathbb{Z}_{N}$ where $N\in\mathbb{N}$ and expresses certain partial sums of coefficients in terms of expressions involving roots of unity. By specializing $N$ to different values, we see that these expressions simplify in some cases and we obtain several nice identities involving these coefficients. We also use a result of Sudler (Quarterly J. Math. 1964) to obtain an asymptotic formula for the maximum absolute value of these coefficients.},

number = {20-08},

year = {2020},

month = {May},

length = {11},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Congruences for generalized Fishburn numbers at roots of unity

#### Ankush Goswami

Technical report no. 20-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2020. [pdf]**techreport**{RISC6119,

author = {Ankush Goswami},

title = {{Congruences for generalized Fishburn numbers at roots of unity}},

language = {english},

abstract = {There has been significant recent interest in the arithmeticproperties of the coefficients of $F(1-q)$ and $\mathcal{F}_t(1-q)$where $F(q)$ is the Kontsevich-Zagier strange series and $\mathcal{F}_t(q)$ is the strange series associated to a family of torus knots as studied by Bijaoui, Boden, Myers, Osburn, Rushworth, Tronsgardand Zhou. In this paper, we prove prime power congruences for two families of generalized Fishburn numbers, namely, for the coefficients of $(\zeta_N - q)^s F((\zeta_N - q)^r)$ and $(\zeta_N - q)^s \mathcal{F}_t((\zeta_N -q)^r)$, where $\zeta_N$ is an $N$th root of unity and $r$, $s$ are certain integers.},

number = {20-09},

year = {2020},

length = {17},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Some structural results on D^n finite functions

#### A. Jimenez-Pastor, V. Pillwein, M.F. Singer

Advances in Applied Mathematics 117, pp. 0-0. June 2020. Elsevier, 0196-8858. [url] [pdf]**article**{RISC6077,

author = {A. Jimenez-Pastor and V. Pillwein and M.F. Singer},

title = {{Some structural results on D^n finite functions}},

language = {english},

abstract = {D-finite (or holonomic) functions satisfy linear differential equations with polynomial coefficients. They form a large class of functions that appear in many applications in Mathematics or Physics. It is well-known that these functions are closed under certain operations and these closure properties can be executed algorithmically. Recently, the notion of D-finite functions has been generalized to differentially definable or Dn-finite functions. Also these functions are closed under operations such as forming (anti)derivative, addition or multiplication and, again, these can be implemented. In this paper we investigate how Dn-finite functions behave under composition and how they are related to algebraic and differentially algebraic functions.},

journal = {Advances in Applied Mathematics},

volume = {117},

pages = {0--0},

publisher = {Elsevier},

isbn_issn = {0196-8858},

year = {2020},

month = {June},

refereed = {yes},

length = {0},

url = {https://doi.org/10.1016/j.aam.2020.102027}

}

### On the exponential generating function of labelled trees

#### Alin Bostan, Antonio Jiménez-Pastor

Comptes Rendus. Mathématique 358(9-10), pp. 1005-1009. 2020. Académie des sciences, Paris, 1631-0721. [url]**article**{RISC6281,

author = {Alin Bostan and Antonio Jiménez-Pastor},

title = {{On the exponential generating function of labelled trees}},

language = {english},

journal = {Comptes Rendus. Mathématique},

volume = {358},

number = {9-10},

pages = {1005--1009},

publisher = {Académie des sciences, Paris},

isbn_issn = {1631-0721},

year = {2020},

refereed = {yes},

length = {5},

url = {https://doi.org/10.5802/crmath.108}

}