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. ...

# Priv.-Doz. Dr. Manuel Kauers

### Research Area

Algorithmic Combinatorics, Computer Algebra## Software

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 ...

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 ...

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, ...

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

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. ...

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 ...

## Publications

### 2016

### Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra

#### Manuel Kauers, Peter Paule, Greg Reid

ACM Communications in Computer Algebra 50(Issue 1), pp. 27-34. March 2016. 1932-2240. [pdf]**article**{RISC5284,

author = {Manuel Kauers and Peter Paule and Greg Reid},

title = {{Workshop on Symbolic Combinatorics and Algorithmic Differential Algebra}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {50},

number = {Issue 1},

pages = {27--34},

isbn_issn = {1932-2240},

year = {2016},

month = {March},

refereed = {no},

length = {8}

}

### 2014

### Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph

#### Manuel Kauers, Ryan ODonnell, Li-Yang Tan, Yuan Zhou

In: Proceedings of SODA'14, tba (ed.), pp. ?-?. 2014. tba. [pdf]**inproceedings**{RISC4829,

author = {Manuel Kauers and Ryan ODonnell and Li-Yang Tan and Yuan Zhou},

title = {{Hypercontractive inequalities via SOS, and the Frankl-R\"odl graph}},

booktitle = {{Proceedings of SODA'14}},

language = {english},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {tba},

refereed = {yes},

length = {15}

}

### Ore Polynomials in Sage

#### Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson

In: Computer Algebra and Polynomials, Jaime Gutierrez, Josef Schicho, Martin Weimann (ed.), Lecture Notes in Computer Science , pp. ?-?. 2014. tba. [pdf] [ps]**inproceedings**{RISC4944,

author = {Manuel Kauers and Maximilian Jaroschek and Fredrik Johansson},

title = {{Ore Polynomials in Sage}},

booktitle = {{Computer Algebra and Polynomials}},

language = {english},

abstract = {We present a Sage implementation of Ore algebras. The main features for the mostcommon instances include basic arithmetic and actions; GCRD and LCLM; D-finiteclosure properties; natural transformations between related algebras; guessing;desingularization; solvers for polynomials, rational functions and (generalized)power series. This paper is a tutorial on how to use the package.},

series = {Lecture Notes in Computer Science},

pages = {?--?},

isbn_issn = {tba},

year = {2014},

editor = {Jaime Gutierrez and Josef Schicho and Martin Weimann},

refereed = {yes},

length = {17}

}

### On the length of integers in telescopers for proper hypergeometric terms

#### Manuel Kauers, Lily Yen

Journal of Symbolic Computation, pp. ?-?. 2014. ISSN 0747-7171. to appear. [pdf] [ps]**article**{RISC4955,

author = {Manuel Kauers and Lily Yen},

title = {{On the length of integers in telescopers for proper hypergeometric terms}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {?--?},

isbn_issn = {ISSN 0747-7171},

year = {2014},

note = {to appear},

refereed = {yes},

length = {15}

}

### Computer Algebra

#### Manuel Kauers

In: Handbook of Combinatorics, Miklos Bona (ed.), pp. ?-?. 2014. Taylor and Francis, tba.**incollection**{RISC4956,

author = {Manuel Kauers},

title = {{Computer Algebra}},

booktitle = {{Handbook of Combinatorics}},

language = {english},

pages = {?--?},

publisher = {Taylor and Francis},

isbn_issn = {tba},

year = {2014},

editor = {Miklos Bona},

refereed = {yes},

length = {59}

}

### Bounds for D-Finite Closure Properties

#### Manuel Kauers

In: Proceedings of ISSAC 2014, Katsusuke Nabeshima (ed.), pp. 288-295. 2014. isbn 978-1-4503-2501-1/14/07. [pdf]**inproceedings**{RISC4989,

author = {Manuel Kauers},

title = {{Bounds for D-Finite Closure Properties}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {288--295},

isbn_issn = {isbn 978-1-4503-2501-1/14/07},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

### A Generalized Apagodu-Zeilberger Algorithm

#### Shaoshi Chen, Manuel Kauers, Christoph Koutschan

In: Proceedings of ISSAC 2014, Katsusuke Nabeshima (ed.), pp. 107-114. 2014. ISBN 978-1-4503-2501-1. [pdf]**inproceedings**{RISC5034,

author = {Shaoshi Chen and Manuel Kauers and Christoph Koutschan},

title = {{A Generalized Apagodu-Zeilberger Algorithm}},

booktitle = {{Proceedings of ISSAC 2014}},

language = {english},

pages = {107--114},

isbn_issn = {ISBN 978-1-4503-2501-1},

year = {2014},

editor = {Katsusuke Nabeshima},

refereed = {yes},

length = {8}

}

### On 3-dimensional lattice walks confined to the positive octant

#### Alin Bostan, Mireille Bousquet-Melou, Manuel Kauers, Stephen Melczer

Annals of Combinatorics, pp. ??-??. 2014. ISSN 0218-0006. to appear. [pdf]**article**{RISC5082,

author = {Alin Bostan and Mireille Bousquet-Melou and Manuel Kauers and Stephen Melczer},

title = {{ On 3-dimensional lattice walks confined to the positive octant}},

language = {english},

journal = {Annals of Combinatorics},

pages = {??--??},

isbn_issn = {ISSN 0218-0006},

year = {2014},

note = {to appear},

refereed = {yes},

length = {36}

}

### 2013

### Formal Laurent Series in Several Variables

#### Ainhoa Aparicio Monforte, Manuel Kauers

Expositiones Mathematicae 31(4), pp. 350-367. 2013. ISSN 0723-0869. [pdf]**article**{RISC4600,

author = {Ainhoa Aparicio Monforte and Manuel Kauers},

title = {{Formal Laurent Series in Several Variables}},

language = {english},

abstract = { We explain the construction of fields of formal infinite series in several variables, generalizing the classical notion of formal Laurent series in one variable. Our discussion addresses the field operations for these series (addition, multiplication, and division), the composition, and includes an implicit function theorem.},

journal = {Expositiones Mathematicae},

volume = {31},

number = {4},

pages = {350--367},

isbn_issn = {ISSN 0723-0869},

year = {2013},

refereed = {yes},

length = {24}

}

### Desingularization Explains Order-Degree Curves for Ore Operators

#### Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, Michael F. Singer

In: Proceedings of ISSAC'13, Manuel Kauers (ed.), pp. 157-164. 2013. isbn 978-1-4503-2059-7/13/06. [pdf] [ps]**inproceedings**{RISC4706,

author = {Shaoshi Chen and Maximilian Jaroschek and Manuel Kauers and Michael F. Singer},

title = {{Desingularization Explains Order-Degree Curves for Ore Operators}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed. An order-degree curve for a given Ore operator is a curve in the $(r,d)$-plane such that for all points $(r,d)$ above this curve, there exists a left multiple of order~$r$ and degree~$d$ of the given operator. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies order-degree curves which are extremely accurate in examples. },

pages = {157--164},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

### Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation

#### Fredrik Johansson, Manuel Kauers, Marc Mezzarobba

In: Proceedings of ISSAC'13, Manuel Kauers (ed.), pp. 211-218. 2013. isbn 978-1-4503-2059-7/13/06. [pdf] [ps]**inproceedings**{RISC4707,

author = {Fredrik Johansson and Manuel Kauers and Marc Mezzarobba},

title = {{Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation}},

booktitle = {{Proceedings of ISSAC'13}},

language = {english},

abstract = { We present a new algorithm for computing hyperexponential solutions of linear ordinary differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions. },

pages = {211--218},

isbn_issn = {isbn 978-1-4503-2059-7/13/06},

year = {2013},

editor = {Manuel Kauers},

refereed = {yes},

length = {8}

}

### The Holonomic Toolbox

#### Manuel Kauers

In: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Johannes Bluemlein and Carsten Schneider (ed.), pp. ??-??. 2013. Springer, tba. to appear. [pdf]**incollection**{RISC4710,

author = {Manuel Kauers},

title = {{The Holonomic Toolbox}},

booktitle = {{Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions}},

language = {english},

abstract = {This is an overview over standard techniques for holonomic functions,written for readers who are new to the subject. We state the definition for holonomyin a couple of different ways, including some concrete special cases as well asa more abstract and more general version. We give a collection of standard examplesand state several fundamental properties of holonomic objects. Two techniqueswhich are most useful in applications are explained in some more detail: closureproperties, which can be used to prove identities among holonomic functions, andguessing, which can be used to generate plausible conjectures for equations satisfiedby a given function.},

pages = {??--??},

publisher = {Springer},

isbn_issn = {tba},

year = {2013},

note = {to appear},

editor = {Johannes Bluemlein and Carsten Schneider},

refereed = {yes},

length = {25}

}

### Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order

#### S. Gerhold, M. Kauers, C. Koutschan, P. Paule, C. Schneider, B. Zimmermann

In: Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, C. Schneider, J. Bluemlein (ed.), Texts and Monographs in Symbolic Computation , pp. 75-96. 2013. Springer, ISBN-13: 978-3709116159. arXiv:1305.4818 [cs.SC]. [url]**incollection**{RISC4721,

author = {S. Gerhold and M. Kauers and C. Koutschan and P. Paule and C. Schneider and B. Zimmermann},

title = {{Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order}},

booktitle = {{Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions}},

language = {english},

series = {Texts and Monographs in Symbolic Computation},

pages = {75--96},

publisher = {Springer},

isbn_issn = {ISBN-13: 978-3709116159},

year = {2013},

note = {arXiv:1305.4818 [cs.SC]},

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

refereed = {no},

length = {22},

url = {http://arxiv.org/abs/1305.4818}

}

### 2012

### Trading Order for Degree in Creative Telescoping

#### Shaoshi Chen, Manuel Kauers

Journal of Symbolic Compuation 47(8), pp. 968-995. 2012. ISSN 0747-7171. [pdf] [ps]**article**{RISC4444,

author = {Shaoshi Chen and Manuel Kauers},

title = {{Trading Order for Degree in Creative Telescoping}},

language = {english},

journal = {Journal of Symbolic Compuation},

volume = {47},

number = {8},

pages = {968--995},

isbn_issn = {ISSN 0747-7171},

year = {2012},

refereed = {yes},

length = {33}

}

### Order-Degree Curves for Hypergeometric Creative Telescoping

#### Shaoshi Chen, Manuel Kauers

In: Proceedings of ISSAC 2012, Joris van der Hoeven and Mark van Hoeij (ed.), pp. 122-129. 2012. isbn 978-1-4503-1269. [pdf]**inproceedings**{RISC4533,

author = {Shaoshi Chen and Manuel Kauers},

title = {{Order-Degree Curves for Hypergeometric Creative Telescoping}},

booktitle = {{Proceedings of ISSAC 2012}},

language = {english},

pages = {122--129},

isbn_issn = {isbn 978-1-4503-1269},

year = {2012},

editor = {Joris van der Hoeven and Mark van Hoeij},

refereed = {yes},

length = {8}

}

### Telescopers for Rational and Algebraic Functions via Residues

#### Shaoshi Chen, Manuel Kauers, Michael F. Singer

In: Proceedings of ISSAC 2012, Joris van der Hoeven and Mark van Hoeij (ed.), pp. 130-137. 2012. isbn 978-1-4503-1269. [pdf]**inproceedings**{RISC4534,

author = {Shaoshi Chen and Manuel Kauers and Michael F. Singer},

title = {{Telescopers for Rational and Algebraic Functions via Residues}},

booktitle = {{Proceedings of ISSAC 2012}},

language = {english},

pages = {130--137},

isbn_issn = {isbn 978-1-4503-1269},

year = {2012},

editor = {Joris van der Hoeven and Mark van Hoeij},

refereed = {yes},

length = {8}

}

### A method for determining the mod-$2^k$ behaviour of recursive sequences, with applications to subgroup counting

#### Manuel Kauers, Christian Krattenthaler, Thomas W. Mueller

Electronic Journal of Combinatorics 18(2), pp. 1-76. 2012. 1077-8926. P37. [pdf] [ps]**article**{RISC4556,

author = {Manuel Kauers and Christian Krattenthaler and Thomas W. Mueller},

title = {{A method for determining the mod-$2^k$ behaviour of recursive sequences, with applications to subgroup counting}},

language = {english},

abstract = {We present a method to obtain congruences modulo powers of $2$ forsequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fu\ss--Catalan numbers, and to subgroup countingfunctions associated with Hecke groups and their lifts.This leads to numerous new results, including many extensionsof known results to higher powers of $2$. },

journal = {Electronic Journal of Combinatorics},

volume = {18},

number = {2},

pages = {1--76},

isbn_issn = {1077-8926},

year = {2012},

note = {P37},

refereed = {yes},

length = {76}

}

### 2011

### The Concrete Tetrahedron

#### Manuel Kauers, Peter Paule

Text and Monographs in Symbolic Computation 1st edition, 2011. Springer Wien, 978-3-7091-0444-6.**book**{RISC4162,

author = {Manuel Kauers and Peter Paule},

title = {{The Concrete Tetrahedron}},

language = {english},

series = {Text and Monographs in Symbolic Computation},

publisher = {Springer Wien},

isbn_issn = {978-3-7091-0444-6},

year = {2011},

edition = {1st},

translation = {0},

length = {210}

}

### Proof of George Andrews's and David Robbins's q-TSPP conjecture

#### Christoph Koutschan, Manuel Kauers, Doron Zeilberger

Proceedings of the National Academy of Sciences 108(6), pp. 2196-2199. 2011. ISSN 0027-8424. [url] [pdf]**article**{RISC4183,

author = {Christoph Koutschan and Manuel Kauers and Doron Zeilberger},

title = {{Proof of George Andrews's and David Robbins's q-TSPP conjecture}},

language = {english},

abstract = {The conjecture that the orbit-counting generating function for totallysymmetric plane partitions can be written as an explicitproduct formula, has been stated independently by George Andrews andDavid Robbins around 1983. We present a proof of this long-standingconjecture.},

journal = {Proceedings of the National Academy of Sciences},

volume = {108},

number = {6},

pages = {2196--2199},

isbn_issn = {ISSN 0027-8424},

year = {2011},

refereed = {yes},

length = {4},

url = {http://www.risc.jku.at/people/ckoutsch/qtspp/}

}

### How To Use Cylindrical Algebraic Decomposition

#### Manuel Kauers

Seminaire Lotharingien de Combinatoire 65(B65a), pp. 1-16. 2011. ISSN 1286-4889. [pdf] [ps]**article**{RISC4231,

author = {Manuel Kauers},

title = {{How To Use Cylindrical Algebraic Decomposition}},

language = {english},

abstract = { We take some items from a textbook on inequalities and show how to prove them with computer algebra using the Cylindrical Algebraic Decomposition algorithm. This is an example collection for standard applications of this algorithm, intended as a guide for potential users.},

journal = {Seminaire Lotharingien de Combinatoire},

volume = {65},

number = {B65a},

pages = {1--16},

isbn_issn = {ISSN 1286-4889},

year = {2011},

refereed = {yes},

length = {16}

}