# Computer Algebra and Combinatorial Inequalities [F050-07]

### Project Lead

### Project Duration

01/03/2013 - 28/02/2021### Project URL

Go to Website## Members

## Veronika Pillwein

## Ali Uncu

## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2019

### On the positivity of the Gillis–Reznick–Zeilberger rational function

#### V. Pillwein

Advances in Applied Mathematics 104, pp. 75 - 84. 2019. ISSN 0196-8858. [url]@

author = {V. Pillwein},

title = {{On the positivity of the Gillis–Reznick–Zeilberger rational function}},

language = {english},

abstract = {In this note we provide further evidence for a conjecture of Gillis, Reznick, and Zeilberger on the positivity of the diagonal coefficients of a multivariate rational function. Kauers had proven this conjecture for up to 6 variables using computer algebra. We present a variation of his approach that allows us to prove positivity of the coefficients up to 17 variables using symbolic computation.},

journal = {Advances in Applied Mathematics},

volume = {104},

pages = {75 -- 84},

isbn_issn = { ISSN 0196-8858},

year = {2019},

refereed = {yes},

keywords = {Positivity, Cylindrical decomposition, Rational function, Symbolic summation},

length = {10},

url = {http://www.sciencedirect.com/science/article/pii/S0196885818301179}

}

**article**{RISC5813,author = {V. Pillwein},

title = {{On the positivity of the Gillis–Reznick–Zeilberger rational function}},

language = {english},

abstract = {In this note we provide further evidence for a conjecture of Gillis, Reznick, and Zeilberger on the positivity of the diagonal coefficients of a multivariate rational function. Kauers had proven this conjecture for up to 6 variables using computer algebra. We present a variation of his approach that allows us to prove positivity of the coefficients up to 17 variables using symbolic computation.},

journal = {Advances in Applied Mathematics},

volume = {104},

pages = {75 -- 84},

isbn_issn = { ISSN 0196-8858},

year = {2019},

refereed = {yes},

keywords = {Positivity, Cylindrical decomposition, Rational function, Symbolic summation},

length = {10},

url = {http://www.sciencedirect.com/science/article/pii/S0196885818301179}

}

### 2018

### On some polynomials and series of Bloch-Polya Type

#### Berkovich A., Uncu A. K.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 146(7), pp. 2827-2838. July 2018. 1088-6826. [url]@

author = {Berkovich A. and Uncu A.~K.},

title = {{On some polynomials and series of Bloch-Polya Type}},

language = {english},

journal = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY},

volume = {146},

number = {7},

pages = {2827--2838},

isbn_issn = {1088-6826},

year = {2018},

month = {July},

refereed = {yes},

keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A17, 05A19, 11B65, 11P81},

length = {12},

url = {http://www.ams.org/journals/proc/2018-146-07/S0002-9939-2018-13982-9/}

}

**article**{RISC5557,author = {Berkovich A. and Uncu A.~K.},

title = {{On some polynomials and series of Bloch-Polya Type}},

language = {english},

journal = {PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY},

volume = {146},

number = {7},

pages = {2827--2838},

isbn_issn = {1088-6826},

year = {2018},

month = {July},

refereed = {yes},

keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A17, 05A19, 11B65, 11P81},

length = {12},

url = {http://www.ams.org/journals/proc/2018-146-07/S0002-9939-2018-13982-9/}

}

### Some Elementary Partition Inequalities and Their Implications

#### Berkovich A., Uncu A. K.

ArXiv e-prints (to appear in Annals of Cobinatorics), pp. -. 2018. Preprint. [url]@

author = {Berkovich A. and Uncu A.~K.},

title = {{Some Elementary Partition Inequalities and Their Implications}},

language = {english},

journal = {ArXiv e-prints (to appear in Annals of Cobinatorics)},

pages = {--},

isbn_issn = {Preprint},

year = {2018},

refereed = {yes},

keywords = {Mathematics - Combinatorics, Mathematics - Number Theory, 05A15, 05A17, 05A19, 05A20, 11B65, 11P81, 11P84, 33D15},

length = {12},

url = {https://arxiv.org/abs/1708.01957}

}

**article**{RISC5558,author = {Berkovich A. and Uncu A.~K.},

title = {{Some Elementary Partition Inequalities and Their Implications}},

language = {english},

journal = {ArXiv e-prints (to appear in Annals of Cobinatorics)},

pages = {--},

isbn_issn = {Preprint},

year = {2018},

refereed = {yes},

keywords = {Mathematics - Combinatorics, Mathematics - Number Theory, 05A15, 05A17, 05A19, 05A20, 11B65, 11P81, 11P84, 33D15},

length = {12},

url = {https://arxiv.org/abs/1708.01957}

}

### Refined q-Trinomial Coefficients and Two Infinite Hierarcies of q-Series Identities

#### Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (submitted), pp. 1-10. 2018. N/A. [url]@

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarcies of q-Series Identities }},

language = {english},

abstract = {We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarcies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

journal = {ArXiv e-prints (submitted)},

pages = {1--10},

isbn_issn = {N/A},

year = {2018},

refereed = {yes},

length = {10},

url = {https://arxiv.org/abs/1810.12048}

}

**article**{RISC5801,author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarcies of q-Series Identities }},

language = {english},

abstract = {We will prove an identity involving refined q-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined q-trinomials in an iterative fashion in the spirit of Bailey chains. One of these two hierarcies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

journal = {ArXiv e-prints (submitted)},

pages = {1--10},

isbn_issn = {N/A},

year = {2018},

refereed = {yes},

length = {10},

url = {https://arxiv.org/abs/1810.12048}

}

### Algorithmic Arithmetics with DD-Finite Functions

#### Jiménez-Pastor Antonio, Pillwein Veronika

In: Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation, Arreche Carlos (ed.), ISSAC '18 , pp. 231-237. 2018. ACM, New York, NY, USA, ISBN 978-1-4503-5550-6. [url]@

author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{Algorithmic Arithmetics with DD-Finite Functions}},

booktitle = {{Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

series = {ISSAC '18},

pages = {231--237},

publisher = {ACM},

address = {New York, NY, USA},

isbn_issn = {ISBN 978-1-4503-5550-6},

year = {2018},

editor = {Arreche Carlos},

refereed = {yes},

keywords = {algorithms, closure properties, holonomic functions},

length = {7},

url = {http://doi.acm.org/10.1145/3208976.3209009}

}

**inproceedings**{RISC5730,author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{Algorithmic Arithmetics with DD-Finite Functions}},

booktitle = {{Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

series = {ISSAC '18},

pages = {231--237},

publisher = {ACM},

address = {New York, NY, USA},

isbn_issn = {ISBN 978-1-4503-5550-6},

year = {2018},

editor = {Arreche Carlos},

refereed = {yes},

keywords = {algorithms, closure properties, holonomic functions},

length = {7},

url = {http://doi.acm.org/10.1145/3208976.3209009}

}

### A Computable Extension for Holonomic Functions: DD-Finite Functions

#### Jiménez-Pastor Antonio, Pillwein Veronika

Journal of Symbolic Computation, pp. -. 2018. ISSN 0747-7171. In Press. [url]@

author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},

language = {english},

abstract = {Differentiably finite (D-finite) formal power series form a large class of useful functions for which a variety of symbolic algorithms exists. Among these methods are several closure properties that can be carried out automatically. We introduce a natural extension of these functions to a larger class of computable objects for which we prove closure properties. These are again algorithmic. This extension can be iterated constructively preserving the closure properties},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {ISSN 0747-7171},

year = {2018},

note = {In Press},

refereed = {yes},

length = {15},

url = {https://doi.org/10.1016/j.jsc.2018.07.002}

}

**article**{RISC5831,author = {Jiménez-Pastor Antonio and Pillwein Veronika},

title = {{A Computable Extension for Holonomic Functions: DD-Finite Functions}},

language = {english},

abstract = {Differentiably finite (D-finite) formal power series form a large class of useful functions for which a variety of symbolic algorithms exists. Among these methods are several closure properties that can be carried out automatically. We introduce a natural extension of these functions to a larger class of computable objects for which we prove closure properties. These are again algorithmic. This extension can be iterated constructively preserving the closure properties},

journal = {Journal of Symbolic Computation},

pages = {--},

isbn_issn = {ISSN 0747-7171},

year = {2018},

note = {In Press},

refereed = {yes},

length = {15},

url = {https://doi.org/10.1016/j.jsc.2018.07.002}

}

### Positivity of the Gillis-Reznick-Zeilberger rational function

#### V. Pillwein

Technical report no. 18-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2018. [pdf]@

author = {V. Pillwein},

title = {{Positivity of the Gillis-Reznick-Zeilberger rational function}},

language = {english},

number = {18-05},

year = {2018},

length = {12},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC5612,author = {V. Pillwein},

title = {{Positivity of the Gillis-Reznick-Zeilberger rational function}},

language = {english},

number = {18-05},

year = {2018},

length = {12},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 2017

### Variation on a theme of Nathan Fine. New weighted partition identities

#### Berkovich Alexander, Uncu Ali K.

J. Number Theory 176, pp. 226-248. 2017. ISSN 0022-314X. [url]@

author = {Berkovich Alexander and Uncu Ali K.},

title = {{Variation on a theme of Nathan Fine. New weighted partition identities}},

language = {english},

journal = {J. Number Theory},

volume = {176},

pages = {226--248},

isbn_issn = { ISSN 0022-314X},

year = {2017},

refereed = {yes},

length = {23},

url = {https://doi.org/10.1016/j.jnt.2016.12.011}

}

**article**{RISC5552,author = {Berkovich Alexander and Uncu Ali K.},

title = {{Variation on a theme of Nathan Fine. New weighted partition identities}},

language = {english},

journal = {J. Number Theory},

volume = {176},

pages = {226--248},

isbn_issn = { ISSN 0022-314X},

year = {2017},

refereed = {yes},

length = {23},

url = {https://doi.org/10.1016/j.jnt.2016.12.011}

}

### Weighted Rogers-Ramanujan partitions and Dyson crank

#### Uncu Ali Kemal

The Ramanujan Journal, pp. -. May 2017. ISSN 1572-9303. [url]@

author = {Uncu Ali Kemal},

title = {{Weighted Rogers--Ramanujan partitions and Dyson crank}},

language = {english},

abstract = {In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts' sum is n is equal to the number of partitions of n with non-negative crank.},

journal = {The Ramanujan Journal},

pages = {--},

isbn_issn = { ISSN 1572-9303},

year = {2017},

month = {May},

refereed = {yes},

length = {0},

url = {https://doi.org/10.1007/s11139-017-9903-8}

}

**article**{RISC5555,author = {Uncu Ali Kemal},

title = {{Weighted Rogers--Ramanujan partitions and Dyson crank}},

language = {english},

abstract = {In this paper, we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results as well as the different statistics with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts' sum is n is equal to the number of partitions of n with non-negative crank.},

journal = {The Ramanujan Journal},

pages = {--},

isbn_issn = { ISSN 1572-9303},

year = {2017},

month = {May},

refereed = {yes},

length = {0},

url = {https://doi.org/10.1007/s11139-017-9903-8}

}

### 2016

### On partitions with fixed number of even-indexed and odd-indexed odd parts

#### Berkovich Alexander, Uncu Ali Kemal

J. Number Theory 167, pp. 7-30. 2016. ISSN 0022-314X. [url]@

author = {Berkovich Alexander and Uncu Ali Kemal},

title = {{On partitions with fixed number of even-indexed and odd-indexed odd parts}},

language = {english},

journal = {J. Number Theory},

volume = {167},

pages = {7--30},

isbn_issn = { ISSN 0022-314X},

year = {2016},

refereed = {yes},

length = {24},

url = {https://doi.org/10.1016/j.jnt.2016.02.031}

}

**article**{RISC5553,author = {Berkovich Alexander and Uncu Ali Kemal},

title = {{On partitions with fixed number of even-indexed and odd-indexed odd parts}},

language = {english},

journal = {J. Number Theory},

volume = {167},

pages = {7--30},

isbn_issn = { ISSN 0022-314X},

year = {2016},

refereed = {yes},

length = {24},

url = {https://doi.org/10.1016/j.jnt.2016.02.031}

}

### New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions

#### Berkovich A., Kemal Uncu A.

ALLADI60 2016: Analytic Number Theory, Modular Forms and q-Hypergeometric Series , pp. -. 2016. 978-3-319-68375-1. [url]@

author = {Berkovich A. and Kemal Uncu A.},

title = {{New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions}},

language = {english},

journal = {ALLADI60 2016: Analytic Number Theory, Modular Forms and q-Hypergeometric Series },

pages = {--},

isbn_issn = {978-3-319-68375-1},

year = {2016},

refereed = {yes},

keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A15, 05A17, 05A19, 11B34, 11B75, 11P81, 11P84, 33D15},

length = {0},

url = {https://link.springer.com/book/10.1007/978-3-319-68376-8}

}

**article**{RISC5556,author = {Berkovich A. and Kemal Uncu A.},

title = {{New Weighted Partition Theorems with the Emphasis on the Smallest Part of Partitions}},

language = {english},

journal = {ALLADI60 2016: Analytic Number Theory, Modular Forms and q-Hypergeometric Series },

pages = {--},

isbn_issn = {978-3-319-68375-1},

year = {2016},

refereed = {yes},

keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 05A15, 05A17, 05A19, 11B34, 11B75, 11P81, 11P84, 33D15},

length = {0},

url = {https://link.springer.com/book/10.1007/978-3-319-68376-8}

}

### A New Witness Identity for $11|p(11n+6)$

#### Peter Paule, Cristian-Silviu Radu

In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series, George E. Andrews, Frank Garvan (ed.), pp. 625-640. 2016. Springer, 2194-1009. [pdf]@

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

title = {{A New Witness Identity for $11|p(11n+6)$}},

booktitle = {{Analytic Number Theory, Modular Forms and q-Hypergeometric Series}},

language = {english},

pages = {625--640},

publisher = {Springer},

isbn_issn = { 2194-1009},

year = {2016},

editor = { George E. Andrews and Frank Garvan},

refereed = {yes},

length = {16}

}

**inproceedings**{RISC5329,author = {Peter Paule and Cristian-Silviu Radu},

title = {{A New Witness Identity for $11|p(11n+6)$}},

booktitle = {{Analytic Number Theory, Modular Forms and q-Hypergeometric Series}},

language = {english},

pages = {625--640},

publisher = {Springer},

isbn_issn = { 2194-1009},

year = {2016},

editor = { George E. Andrews and Frank Garvan},

refereed = {yes},

length = {16}

}

### 2015

### A new companion to Capparelli's identities

#### Berkovich Alexander, Uncu Ali Kemal

Adv. in Appl. Math. 71, pp. 125-137. 2015. ISSN 0196-8858. [url]@

author = {Berkovich Alexander and Uncu Ali Kemal},

title = {{A new companion to Capparelli's identities}},

language = {english},

journal = {Adv. in Appl. Math.},

volume = {71},

pages = {125--137},

isbn_issn = { ISSN 0196-8858},

year = {2015},

refereed = {yes},

length = {13},

url = {https://doi.org/10.1016/j.aam.2015.09.012}

}

**article**{RISC5554,author = {Berkovich Alexander and Uncu Ali Kemal},

title = {{A new companion to Capparelli's identities}},

language = {english},

journal = {Adv. in Appl. Math.},

volume = {71},

pages = {125--137},

isbn_issn = { ISSN 0196-8858},

year = {2015},

refereed = {yes},

length = {13},

url = {https://doi.org/10.1016/j.aam.2015.09.012}

}

### Proof of a conjecture of M. Patrick concerning Jacobi polynomials

#### A. Alexandrov, H. Dietert, G. Nikolov, V. Pillwein

Journal of Mathematical Analysis and Applications 428(2), pp. 750-761. 2015.@

author = {A. Alexandrov and H. Dietert and G. Nikolov and V. Pillwein},

title = {{Proof of a conjecture of M. Patrick concerning Jacobi polynomials }},

language = {english},

journal = {Journal of Mathematical Analysis and Applications},

volume = {428},

number = {2},

pages = {750--761},

isbn_issn = {?},

year = {2015},

refereed = {yes},

length = {12}

}

**article**{RISC5140,author = {A. Alexandrov and H. Dietert and G. Nikolov and V. Pillwein},

title = {{Proof of a conjecture of M. Patrick concerning Jacobi polynomials }},

language = {english},

journal = {Journal of Mathematical Analysis and Applications},

volume = {428},

number = {2},

pages = {750--761},

isbn_issn = {?},

year = {2015},

refereed = {yes},

length = {12}

}

### Symbolic Computation and Finite Element Methods.

#### V. Pillwein

In: CASC 2015, V.P. Gerdt, W. Koepf, W.M. Seiler, and E.V. Vorozhtsov (ed.), LNCS 9301, pp. 374-388. 2015. Springer-Verlag Berlin Heidelberg, 0302-9743. [pdf]@

author = {V. Pillwein},

title = {{Symbolic Computation and Finite Element Methods.}},

booktitle = {{CASC 2015}},

language = {english},

series = {LNCS},

volume = {9301},

pages = {374--388},

publisher = {Springer-Verlag Berlin Heidelberg},

isbn_issn = {0302-9743},

year = {2015},

editor = {V.P. Gerdt and W. Koepf and W.M. Seiler and and E.V. Vorozhtsov},

refereed = {no},

length = {15}

}

**inproceedings**{RISC5182,author = {V. Pillwein},

title = {{Symbolic Computation and Finite Element Methods.}},

booktitle = {{CASC 2015}},

language = {english},

series = {LNCS},

volume = {9301},

pages = {374--388},

publisher = {Springer-Verlag Berlin Heidelberg},

isbn_issn = {0302-9743},

year = {2015},

editor = {V.P. Gerdt and W. Koepf and W.M. Seiler and and E.V. Vorozhtsov},

refereed = {no},

length = {15}

}

### An efficient procedure deciding positivity for a class of holonomic sequences

#### V. Pillwein, M. Schussler

ACM Communications in Computer Algebra 49(3), pp. 90-93. 2015. Extended abstract of the poster presentation at ISSAC 2015. [pdf]@

author = {V. Pillwein and M. Schussler},

title = {{An efficient procedure deciding positivity for a class of holonomic sequences}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {49},

number = {3},

pages = {90--93},

isbn_issn = { },

year = {2015},

note = {Extended abstract of the poster presentation at ISSAC 2015},

refereed = {yes},

length = {4}

}

**article**{RISC5617,author = {V. Pillwein and M. Schussler},

title = {{An efficient procedure deciding positivity for a class of holonomic sequences}},

language = {english},

journal = {ACM Communications in Computer Algebra},

volume = {49},

number = {3},

pages = {90--93},

isbn_issn = { },

year = {2015},

note = {Extended abstract of the poster presentation at ISSAC 2015},

refereed = {yes},

length = {4}

}

### 2012

### Linear Diophantine Systems: Partition Analysis and Polyhedral Geometry

#### Zafeirakis Zafeirakopoulos

Research Institute for Symbolic Computation / DK-compmath. PhD Thesis. December 2012. [pdf]@

author = {Zafeirakis Zafeirakopoulos},

title = {{Linear Diophantine Systems: Partition Analysis and Polyhedral Geometry}},

language = {english},

year = {2012},

month = {December},

translation = {0},

school = {Research Institute for Symbolic Computation / DK-compmath},

length = {0}

}

**phdthesis**{RISC4715,author = {Zafeirakis Zafeirakopoulos},

title = {{Linear Diophantine Systems: Partition Analysis and Polyhedral Geometry}},

language = {english},

year = {2012},

month = {December},

translation = {0},

school = {Research Institute for Symbolic Computation / DK-compmath},

length = {0}

}