# Computer Algebra Tools for Special Functions [DK6]

### Project Lead

### Project Duration

01/10/2011 - 30/06/2022### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Software

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 ...

Authors: Nicolas Smoot

More## Publications

### 2021

[Smoot]

### A method of verifying partition congruences by symbolic computation

#### Nicolas Allen Smoot, Cristian-Silviu Radu

Journal of Symbolic Computation 104, pp. 105-133. 2021. 0747-7171. [doi] [pdf]@

author = {Nicolas Allen Smoot and Cristian-Silviu Radu},

title = {{A method of verifying partition congruences by symbolic computation}},

language = {english},

abstract = {Conjectures involving infinite families of restricted partition congruences can be difficult to verify for a number of individual cases, even with a computer. We demonstrate how the machinery of Radu's algorithm may be modified and employed to efficiently check a very large number of cases of such conjectures. This allows substantial evidence to be collected for a given conjecture, before a complete proof is attempted.},

journal = {Journal of Symbolic Computation},

volume = {104},

pages = {105--133},

isbn_issn = {0747-7171},

year = {2021},

refereed = {yes},

length = {29},

url = {https://doi.org/10.1016/j.jsc.2020.04.008}

}

**article**{RISC6106,author = {Nicolas Allen Smoot and Cristian-Silviu Radu},

title = {{A method of verifying partition congruences by symbolic computation}},

language = {english},

abstract = {Conjectures involving infinite families of restricted partition congruences can be difficult to verify for a number of individual cases, even with a computer. We demonstrate how the machinery of Radu's algorithm may be modified and employed to efficiently check a very large number of cases of such conjectures. This allows substantial evidence to be collected for a given conjecture, before a complete proof is attempted.},

journal = {Journal of Symbolic Computation},

volume = {104},

pages = {105--133},

isbn_issn = {0747-7171},

year = {2021},

refereed = {yes},

length = {29},

url = {https://doi.org/10.1016/j.jsc.2020.04.008}

}

### 2020

[Smoot]

### Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences

#### N. Smoot

Research Institute for Symbolic Computation, JKU Linz. PhD Thesis. 2020. [pdf]@

author = {N. Smoot},

title = {{Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences}},

language = {english},

abstract = {We give the implementation of an algorithm developed by Silviu Radu to compute examples of a wide variety of arithmetic identities originally studied by Ramanujan and Kolberg. Such identities employ certain finiteness conditions imposed by the theory of modular functions, and often yield interesting arithmetic information about the integer partition function $p(n)$, and other associated functions. We compute a large number of examples of such identities taken from contemporary research, often extending or improving existing results. We then use our implementation as a computational tool to help us achieve more theoretical results in the study of infinite congruence families. We finally describe a new method which extends the existing techniques for proving partition congruence families associated with a genus 0 modular curve.},

year = {2020},

translation = {0},

school = {Research Institute for Symbolic Computation, JKU Linz},

length = {224}

}

**phdthesis**{RISC6512,author = {N. Smoot},

title = {{Computer Algebra with the Fifth Operation: Applications of Modular Functions to Partition Congruences}},

language = {english},

abstract = {We give the implementation of an algorithm developed by Silviu Radu to compute examples of a wide variety of arithmetic identities originally studied by Ramanujan and Kolberg. Such identities employ certain finiteness conditions imposed by the theory of modular functions, and often yield interesting arithmetic information about the integer partition function $p(n)$, and other associated functions. We compute a large number of examples of such identities taken from contemporary research, often extending or improving existing results. We then use our implementation as a computational tool to help us achieve more theoretical results in the study of infinite congruence families. We finally describe a new method which extends the existing techniques for proving partition congruence families associated with a genus 0 modular curve.},

year = {2020},

translation = {0},

school = {Research Institute for Symbolic Computation, JKU Linz},

length = {224}

}

### 2019

[Hemmecke]

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

#### Ralf Hemmecke, Silviu Radu

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. [doi]@

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},

note = {Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf},

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}

}

**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},

note = {Also available as RISC Report 18-03 http://www.risc.jku.at/publications/download/risc_5561/etarelations.pdf},

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, Johannes Blümlein and Carsten Schneider and Peter Paule (ed.), 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. [doi]@

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},

address = {Cham},

isbn_issn = {978-3-030-04479-4},

year = {2019},

note = {Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf},

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}

}

**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},

address = {Cham},

isbn_issn = {978-3-030-04479-4},

year = {2019},

note = {Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf},

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}

}

### 2017

[Ye]

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

#### Liangjie Ye

Journal of Symbolic Computation, to appear, pp. 1-25. 2017. -. [pdf]@

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}

}

**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]@

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}

}

**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]@

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}

}

**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}

}

### 2015

[Breuer]

### Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems

#### Felix Breuer, Zafeirakis Zafeirakopoulos

Technical report no. 15-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2015. [pdf]@

author = {Felix Breuer and Zafeirakis Zafeirakopoulos},

title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},

language = {english},

abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},

number = {15-09},

year = {2015},

month = {January},

keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},

length = {49},

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)}

}

**techreport**{RISC5153,author = {Felix Breuer and Zafeirakis Zafeirakopoulos},

title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},

language = {english},

abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},

number = {15-09},

year = {2015},

month = {January},

keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},

length = {49},

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)}

}

### 2011

[Ye]

### Lower Bounds and Constructions for q-ary Codes Correcting Asymmetric Errors

#### Qunying Liao, Liangjie Ye

Advances in Mathematics(China) 42(6), pp. 795-800. 2011. ISSN:1000-0917 . [url] [pdf]@

author = {Qunying Liao and Liangjie Ye},

title = {{Lower Bounds and Constructions for q-ary Codes Correcting Asymmetric Errors}},

language = {english},

journal = {Advances in Mathematics(China)},

volume = {42},

number = {6},

pages = {795--800},

isbn_issn = {ISSN:1000-0917 },

year = {2011},

refereed = {yes},

length = {6},

url = {http://advmath.pku.edu.cn/EN/volumn/volumn_1337.shtml}

}

**article**{RISC4569,author = {Qunying Liao and Liangjie Ye},

title = {{Lower Bounds and Constructions for q-ary Codes Correcting Asymmetric Errors}},

language = {english},

journal = {Advances in Mathematics(China)},

volume = {42},

number = {6},

pages = {795--800},

isbn_issn = {ISSN:1000-0917 },

year = {2011},

refereed = {yes},

length = {6},

url = {http://advmath.pku.edu.cn/EN/volumn/volumn_1337.shtml}

}