# Partition Analysis [F050-06]

### Project Duration

01/03/2013 - 28/02/2021

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## Publications

[Goswami]

### A q-analogue for Euler’s evaluations of the Riemann zeta function

#### Ankush Goswami

Research in Number Theory 5:3, pp. 1-11. 2019. Springer, 10.1007.
@article{RISC5959,
author = {Ankush Goswami},
title = {{A q-analogue for Euler’s evaluations of the Riemann zeta function}},
language = {english},
journal = {Research in Number Theory},
volume = {5:3},
pages = {1--11},
publisher = {Springer},
isbn_issn = {10.1007},
year = {2019},
refereed = {yes},
length = {11}
}
[Goswami]

### A q-analogue for Euler’s $\zeta(6)=\pi^6/6$

#### Ankush Goswami

In: Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, , Proceedings of Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday, pp. 1-5. 2019. none.
@inproceedings{RISC5960,
author = {Ankush Goswami},
title = {{A q-analogue for Euler’s $\zeta(6)=\pi^6/6$}},
booktitle = {{Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}},
language = {english},
pages = {1--5},
isbn_issn = {none},
year = {2019},
editor = {Springer-Birkhauser},
refereed = {yes},
length = {5},
conferencename = {Combinatory Analysis 2018: A Conference in Honor of George Andrews' 80th Birthday}
}
[Goswami]

### Some Problems in Analytic Number Theory

#### Ankush Goswami

University of Florida. PhD Thesis. 2019. First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan - A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY.
@phdthesis{RISC5961,
author = {Ankush Goswami},
title = {{Some Problems in Analytic Number Theory}},
language = {english},
year = {2019},
note = {First part of this thesis is to appear in Proceedings of Analytic and Combinatorial Number Theory: The Legacy of Ramanujan -- A CONFERENCE IN HONOR OF BRUCE C. BERNDT'S 80TH BIRTHDAY},
translation = {0},
school = {University of Florida},
length = {90}
}
[Hemmecke]

### Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$

Journal of Symbolic Compuation 95, pp. 39-52. 2019. ISSN 0747-7171. Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf. [url]
@article{RISC5703,
author = {Ralf Hemmecke and Silviu Radu},
title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},
language = {english},
abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind$\eta$-functions of level $N$, i.e., among the elements of$\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where$1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of$N$.More precisely, we find a finite generating set (which is also aGr\"obner basis of the ideal $\ker\phi$ where\begin{gather*} \phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},
journal = {Journal of Symbolic Compuation},
volume = {95},
pages = {39--52},
isbn_issn = {ISSN 0747-7171},
year = {2019},
refereed = {yes},
keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},
length = {14},
url = {https://doi.org/10.1016/j.jsc.2018.10.001}
}
[Hemmecke]

### The Generators of all Polynomial Relations among Jacobi Theta Functions

#### Ralf Hemmecke, Silviu Radu, Liangjie Ye

In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, , Texts & Monographs in Symbolic Computation 18-09, pp. 259-268. 2019. Springer International Publishing, Cham, 978-3-030-04479-4. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf. [url]
@incollection{RISC5913,
author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},
title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},
booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},
language = {english},
abstract = {In this article, we consider the classical Jacobi theta functions$\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},
series = {Texts & Monographs in Symbolic Computation},
number = {18-09},
pages = {259--268},
publisher = {Springer International Publishing},
isbn_issn = {978-3-030-04479-4},
year = {2019},
editor = {Johannes Blümlein and Carsten Schneider and Peter Paule},
refereed = {yes},
length = {9},
url = {https://doi.org/10.1007/978-3-030-04480-0_11}
}
[Hemmecke]

### Construction of Modular Function Bases for $\Gamma_0(121)$ related to $p(11n+6)$

#### Ralh Hemmecke, Peter Paule, Silviu Radu

Technical report no. 19-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. October 2019. [url] [pdf]
@techreport{RISC5983,
author = {Ralh Hemmecke and Peter Paule and Silviu Radu},
title = {{Construction of Modular Function Bases for $\Gamma_0(121)$ related to $p(11n+6)$}},
language = {english},
number = {19-10},
year = {2019},
month = {October},
keywords = {Ramanujan identities, bases for modular functions, integral bases},
length = {16},
url = {https://risc.jku.at/people/hemmecke/papers/integralbasis/},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

### Infinite Product Formulae for Generating Functions for Sequences of Squares

#### Christian Krattenthaler, Mircea Merca, Cristian-Silviu Radu

2019. [pdf]
@techreport{RISC6013,
author = {Christian Krattenthaler and Mircea Merca and Cristian-Silviu Radu},
title = {{Infinite Product Formulae for Generating Functions for Sequences of Squares}},
language = {english},
year = {2019},
length = {37},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

[Jiu]

### The Method of Brackets in Experimental Mathematics

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

In: Frontiers in Orthogonal Polynomials and q-Series, , pp. -. 2018. World Scientific Publishing, 978-981-3228-87-0. [url]
@incollection{RISC5497,
author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},
title = {{The Method of Brackets in Experimental Mathematics}},
booktitle = {{Frontiers in Orthogonal Polynomials and q-Series}},
language = {english},
pages = {--},
publisher = {World Scientific Publishing},
isbn_issn = {978-981-3228-87-0},
year = {2018},
editor = {Xin Li and Zuhair Nashed},
refereed = {no},
length = {0},
url = {http://www.worldscientific.com/worldscibooks/10.1142/10677}
}
[Magnusson]

### The functional equation of Dedekind's $\eta$-function

#### Tobias Magnusson

June 15 2018. [pdf] [tex]
@techreport{RISC5701,
author = {Tobias Magnusson},
title = {{The functional equation of Dedekind's $\eta$-function}},
language = {English},
year = {2018},
month = {June 15},
length = {14},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

[Jiu]

### Integral representations of equally positive integer-indexed harmonic sums at infinity

#### L. Jiu

Research in Number Theory 3(10), pp. 1-4. 2017. 2363-9555. [url]
@article{RISC5385,
author = {L. Jiu},
title = {{Integral representations of equally positive integer-indexed harmonic sums at infinity}},
language = {English},
abstract = {We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.},
journal = {Research in Number Theory},
volume = {3},
number = {10},
pages = {1--4},
isbn_issn = {2363-9555},
year = {2017},
refereed = {no},
length = {4},
url = {https://resnumtheor.springeropen.com/articles/10.1007/s40993-017-0074-x}
}
[Jiu]

### An extension of the method of brackets. Part 1

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

Open Mathematics (formerly Central European Journal of Mathematics) 15, pp. 1181-1211. 2017. 2391-5455. [url]
@article{RISC5483,
author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},
title = {{An extension of the method of brackets. Part 1}},
language = {english},
abstract = {The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_n$ have meromorphic representations for $n\in\mathbb{C}$, but might vanish or blow up when $n\in\mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.},
journal = {Open Mathematics (formerly Central European Journal of Mathematics)},
volume = {15},
pages = {1181--1211},
isbn_issn = {2391-5455},
year = {2017},
refereed = {no},
length = {31},
url = {https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0100/math-2017-0100.xml?format=INT}
}
[Xiong]

### Overpartitions and ternary quadratic forms

#### xinhua xiong

The Ramanujan Journal 42(2), pp. 429-442. 2017. issn:1382-4090.
@article{RISC5417,
author = {xinhua xiong},
title = {{Overpartitions and ternary quadratic forms}},
language = {english},
journal = {The Ramanujan Journal},
volume = {42},
number = {2},
pages = {429--442},
isbn_issn = {issn:1382-4090},
year = {2017},
refereed = {yes},
length = {13}
}
[Ye]

### Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations

#### Liangjie Ye

Journal of Symbolic Computation, to appear, pp. 1-25. 2017. -. [pdf]
@article{RISC5286,
author = {Liangjie Ye},
title = {{Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations}},
language = {english},
journal = {Journal of Symbolic Computation, to appear},
pages = {1--25},
isbn_issn = {-},
year = {2017},
refereed = {yes},
length = {25}
}
[Ye]

### A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions

#### Liangjie Ye

Journal of Symbolic Computation 82, pp. 134-163. 2017. ISSN: 0747-7171. [pdf]
@article{RISC5455,
author = {Liangjie Ye},
title = {{A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions}},
language = {english},
journal = {Journal of Symbolic Computation },
volume = {82},
pages = {134--163},
isbn_issn = {ISSN: 0747-7171},
year = {2017},
refereed = {yes},
length = {30}
}
[Ye]

### Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities

#### Liangjie Ye

RISC and the DK program Linz. PhD Thesis. 2017. Updated version in June 2017. [pdf]
@phdthesis{RISC5463,
author = {Liangjie Ye},
title = {{Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities}},
language = {english},
year = {2017},
note = {Updated version in June 2017},
translation = {0},
school = {RISC and the DK program Linz},
length = {122}
}

[Kronholm]

### A Polyhedral Model of Partitions with Bounded Differences and a Bijective Proof of a Theorem of Andrews, Beck, and Robbins

#### Brandt Kronholm, Felix Breuer

Research in Number Theory, pp. 1-15. March 2016. Springer, 2363-9555.
@article{RISC5268,
author = {Brandt Kronholm and Felix Breuer},
title = {{A Polyhedral Model of Partitions with Bounded Differences and a Bijective Proof of a Theorem of Andrews, Beck, and Robbins}},
language = {english},
journal = {Research in Number Theory},
pages = {1--15},
publisher = {Springer},
isbn_issn = {2363-9555},
year = {2016},
month = {March},
refereed = {yes},
length = {15}
}
[Paule]

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

In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series, , pp. 625-640. 2016. Springer, 2194-1009. [pdf]
@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}
}
[Xiong]

### Overpartition function modulo 16 and some binary quadratic forms

#### Xinhua Xiong

Technical report no. 5 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. ISSN: 1793-0421, 8 2016.
@techreport{RISC5309,
author = {Xinhua Xiong},
title = {{Overpartition function modulo 16 and some binary quadratic forms}},
language = {english},
number = {5},
isbn_issn = { ISSN: 1793-0421},
year = {2016},
month = {8},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Xiong]

### Overpartition function modulo 16 and some binary quadratic forms

#### Xinhua Xiong

International Journal of Number Theory 12(5), pp. 1195-1208. 2016. ISSN 1793-0421.
@article{RISC5310,
author = {Xinhua Xiong},
title = {{Overpartition function modulo 16 and some binary quadratic forms}},
language = {english},
journal = {International Journal of Number Theory},
volume = {12},
number = {5},
pages = {1195--1208},
isbn_issn = {ISSN 1793-0421},
year = {2016},
refereed = {yes},
length = {13}
}
[Xiong]

### A positivity conjecture related first positive rank and crank moments for overpartitions

#### Xinhua Xiong

Proc. Japan Acad. Ser. A: Math. Sci., 92 (2016), no. 11,, pp. 117-120. 2016. ISSN . [url]
@article{RISC5311,
author = {Xinhua Xiong},
title = {{A positivity conjecture related first positive rank and crank moments for overpartitions}},
language = {english},
journal = { Proc. Japan Acad. Ser. A: Math. Sci., 92 (2016), no. 11,},
pages = {117--120},
isbn_issn = {ISSN },
year = {2016},
refereed = {yes},
length = {6},
url = {http://arxiv.org/abs/1605.09135}
}