## Members

## Publications

### 2020

[Cerna]

### Unital Anti-Unification: Type and Algorithms

#### David M. Cerna , Temur Kutsia

Technical report no. 20-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. RISC Report, Febrary 2020. [pdf]@

author = {David M. Cerna and Temur Kutsia},

title = {{Unital Anti-Unification: Type and Algorithms}},

language = {english},

abstract = {Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete and return tree grammars from which set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions. },

number = {20-02},

year = {2020},

month = {Febrary},

howpublished = {RISC Report},

keywords = {Anti-unification, tree grammars, unital theories, collapse theories},

length = {19},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC6080,author = {David M. Cerna and Temur Kutsia},

title = {{Unital Anti-Unification: Type and Algorithms}},

language = {english},

abstract = {Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete and return tree grammars from which set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions. },

number = {20-02},

year = {2020},

month = {Febrary},

howpublished = {RISC Report},

keywords = {Anti-unification, tree grammars, unital theories, collapse theories},

length = {19},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

[Kutsia]

### Unification modulo alpha-equivalence in a mathematical assistant system

#### Temur Kutsia

Technical report no. 20-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2020. [pdf]@

author = {Temur Kutsia},

title = {{Unification modulo alpha-equivalence in a mathematical assistant system}},

language = {english},

number = {20-01},

year = {2020},

length = {21},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC6074,author = {Temur Kutsia},

title = {{Unification modulo alpha-equivalence in a mathematical assistant system}},

language = {english},

number = {20-01},

year = {2020},

length = {21},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

[Paule]

### Holonomic Relations for Modular Functions and Forms: First Guess, then Prove

#### Peter Paule, Silviu Radu

2020. [pdf]@

author = {Peter Paule and Silviu Radu},

title = {{Holonomic Relations for Modular Functions and Forms: First Guess, then Prove}},

language = {english},

year = {2020},

length = {46},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

**techreport**{RISC6081,author = {Peter Paule and Silviu Radu},

title = {{Holonomic Relations for Modular Functions and Forms: First Guess, then Prove}},

language = {english},

year = {2020},

length = {46},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

[Pillwein]

### A sequence of polynomials generated by a Kapteyn series of the second kind

#### D. Dominici, V. Pillwein

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, V. Pillwein and C. Schneider (ed.), Texts and Monographs in Symbolic Computuation, in press , pp. ?-?. 2020. Springer, arXiv:1607.05314 [math.CO]. [url] [pdf]@

author = {D. Dominici and V. Pillwein},

title = {{A sequence of polynomials generated by a Kapteyn series of the second kind}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation, in press},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2020},

note = {arXiv:1607.05314 [math.CO]},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {0},

url = {https://www.dk-compmath.jku.at/publications/dk-reports/2019-05-28/view}

}

**incollection**{RISC6078,author = {D. Dominici and V. Pillwein},

title = {{A sequence of polynomials generated by a Kapteyn series of the second kind}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation, in press},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2020},

note = {arXiv:1607.05314 [math.CO]},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {0},

url = {https://www.dk-compmath.jku.at/publications/dk-reports/2019-05-28/view}

}

[Schicho]

### Probabilities of incidence between lines and a plane curve over finite field

#### M. Gallet, M. Makhul, J. Schicho

Finite Fields and Their Applications 61, pp. 1-22. 2020. 1071-5797.@

author = {M. Gallet and M. Makhul and J. Schicho},

title = {{Probabilities of incidence between lines and a plane curve over finite field}},

language = {english},

journal = {Finite Fields and Their Applications},

volume = {61},

pages = {1--22},

isbn_issn = {1071-5797},

year = {2020},

refereed = {yes},

length = {22}

}

**article**{RISC6073,author = {M. Gallet and M. Makhul and J. Schicho},

title = {{Probabilities of incidence between lines and a plane curve over finite field}},

language = {english},

journal = {Finite Fields and Their Applications},

volume = {61},

pages = {1--22},

isbn_issn = {1071-5797},

year = {2020},

refereed = {yes},

length = {22}

}

[Schneider]

### Evaluation of binomial double sums involving absolute values

#### C. Krattenthaler, C. Schneider

In: Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday, V. Pillwein, C. Schneider (ed.), Texts and Monographs in Symbolic Computuation, in press , pp. ?-?. 2020. Springer, arXiv:1607.05314 [math.CO]. [url]@

author = {C. Krattenthaler and C. Schneider},

title = {{Evaluation of binomial double sums involving absolute values}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation, in press},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2020},

note = {arXiv:1607.05314 [math.CO]},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1607.05314}

}

**incollection**{RISC5970,author = {C. Krattenthaler and C. Schneider},

title = {{Evaluation of binomial double sums involving absolute values}},

booktitle = {{Algorithmic Combinatorics: Enumerative Combinatorics, Special Functions and Computer Algebra, in Honour of Peter Paule on his 60th Birthday}},

language = {english},

series = {Texts and Monographs in Symbolic Computuation, in press},

pages = {?--?},

publisher = {Springer},

isbn_issn = {?},

year = {2020},

note = {arXiv:1607.05314 [math.CO]},

editor = {V. Pillwein and C. Schneider},

refereed = {yes},

length = {36},

url = {https://arxiv.org/abs/1607.05314}

}

[Schneider]

### Minimal representations and algebraic relations for single nested products

#### C. Schneider

Programming and Computer Software, in press 46(2), pp. ?-?. 2020. ISSN 1608-3261. arXiv:1911.04837 [cs.SC]. [url]@

author = {C. Schneider},

title = {{Minimal representations and algebraic relations for single nested products}},

language = {english},

journal = {Programming and Computer Software, in press},

volume = {46},

number = {2},

pages = {?--?},

isbn_issn = {ISSN 1608-3261},

year = {2020},

note = {arXiv:1911.04837 [cs.SC]},

refereed = {yes},

length = {0},

url = {https://arxiv.org/abs/1911.04837}

}

**article**{RISC6002,author = {C. Schneider},

title = {{Minimal representations and algebraic relations for single nested products}},

language = {english},

journal = {Programming and Computer Software, in press},

volume = {46},

number = {2},

pages = {?--?},

isbn_issn = {ISSN 1608-3261},

year = {2020},

note = {arXiv:1911.04837 [cs.SC]},

refereed = {yes},

length = {0},

url = {https://arxiv.org/abs/1911.04837}

}

[Schneider]

### Three loop QCD corrections to heavy quark form factors

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

In: Proceedings of ACAT 2019, to appear, pp. -. 2020. arXiv:1905.03728 [hep-ph]. [url]@

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

title = {{Three loop QCD corrections to heavy quark form factors}},

booktitle = {{Proceedings of ACAT 2019}},

language = {english},

volume = {to appear},

pages = {--},

isbn_issn = {?},

year = {2020},

note = {arXiv:1905.03728 [hep-ph]},

editor = {?},

refereed = {no},

length = {9},

url = {https://arxiv.org/abs/1905.03728}

}

**inproceedings**{RISC6004,author = {J. Ablinger and J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Three loop QCD corrections to heavy quark form factors}},

booktitle = {{Proceedings of ACAT 2019}},

language = {english},

volume = {to appear},

pages = {--},

isbn_issn = {?},

year = {2020},

note = {arXiv:1905.03728 [hep-ph]},

editor = {?},

refereed = {no},

length = {9},

url = {https://arxiv.org/abs/1905.03728}

}

[Schneider]

### From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms

#### J. Blümlein, A. Maier, P. Marquard, G. Schäfer, C. Schneider

Physics Letters B 801(135157), pp. 1-8. 2020. ISSN 0370-2693. arXiv:1911.04411 [gr-qc]. [url]@

author = {J. Blümlein and A. Maier and P. Marquard and G. Schäfer and C. Schneider},

title = {{From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms}},

language = {english},

journal = {Physics Letters B},

volume = {801},

number = {135157},

pages = {1--8},

isbn_issn = {ISSN 0370-2693},

year = {2020},

note = {arXiv:1911.04411 [gr-qc]},

refereed = {yes},

length = {8},

url = {https://doi.org/10.1016/j.physletb.2019.135157}

}

**article**{RISC6009,author = {J. Blümlein and A. Maier and P. Marquard and G. Schäfer and C. Schneider},

title = {{From Momentum Expansions to Post-Minkowskian Hamiltonians by Computer Algebra Algorithms}},

language = {english},

journal = {Physics Letters B},

volume = {801},

number = {135157},

pages = {1--8},

isbn_issn = {ISSN 0370-2693},

year = {2020},

note = {arXiv:1911.04411 [gr-qc]},

refereed = {yes},

length = {8},

url = {https://doi.org/10.1016/j.physletb.2019.135157}

}

[Schneider]

### Heavy quark form factors at three loops

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

In: 14th International Symposium on Radiative Corrections (RADCOR2019), D. Kosower, M. Cacciari (ed.)POS(RADCOR2019)013, pp. 1-7. 2020. ISSN 1824-8039. [url]@

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

title = {{Heavy quark form factors at three loops}},

booktitle = {{14th International Symposium on Radiative Corrections (RADCOR2019)}},

language = {english},

volume = {POS(RADCOR2019)013},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2020},

editor = {D. Kosower and M. Cacciari},

refereed = {yes},

length = {7},

url = {https://doi.org/10.22323/1.375.0013 }

}

**inproceedings**{RISC6014,author = {J. Blümlein and P. Marquard and N. Rana and C. Schneider},

title = {{Heavy quark form factors at three loops}},

booktitle = {{14th International Symposium on Radiative Corrections (RADCOR2019)}},

language = {english},

volume = {POS(RADCOR2019)013},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2020},

editor = {D. Kosower and M. Cacciari},

refereed = {yes},

length = {7},

url = {https://doi.org/10.22323/1.375.0013 }

}

[Schneider]

### A refined machinery to calculate large moments from coupled systems of linear differential equations

#### Johannes Blümlein, Peter Marquard, Carsten Schneider

In: 14th International Symposium on Radiative Corrections (RADCOR2019), D. Kosower, M. Cacciari (ed.), POS(RADCOR2019)078 , pp. 1-13. 2020. ISSN 1824-8039. arXiv:1912.04390 [cs.SC]. [url]@

author = {Johannes Blümlein and Peter Marquard and Carsten Schneider},

title = {{A refined machinery to calculate large moments from coupled systems of linear differential equations}},

booktitle = {{14th International Symposium on Radiative Corrections (RADCOR2019)}},

language = {english},

series = {POS(RADCOR2019)078},

pages = {1--13},

isbn_issn = {ISSN 1824-8039},

year = {2020},

note = {arXiv:1912.04390 [cs.SC]},

editor = {D. Kosower and M. Cacciari},

refereed = {yes},

length = {13},

url = {https://doi.org/10.22323/1.375.0078}

}

**inproceedings**{RISC6015,author = {Johannes Blümlein and Peter Marquard and Carsten Schneider},

title = {{A refined machinery to calculate large moments from coupled systems of linear differential equations}},

booktitle = {{14th International Symposium on Radiative Corrections (RADCOR2019)}},

language = {english},

series = {POS(RADCOR2019)078},

pages = {1--13},

isbn_issn = {ISSN 1824-8039},

year = {2020},

note = {arXiv:1912.04390 [cs.SC]},

editor = {D. Kosower and M. Cacciari},

refereed = {yes},

length = {13},

url = {https://doi.org/10.22323/1.375.0078}

}

[Schneider]

### The Polarized Three-Loop Anomalous Dimensions from a Massive Calculation

#### A. Behring, J. Blümlein, A. De Freitas, A. Goedicke, S. Klein, A. van Manteuffel, C. Schneider, K. Schönwald

In: 14th International Symposium on Radiative Corrections (RADCOR2019), D. Kosower, M. Cacciari (ed.), POS(RADCOR2019)047 arXiv:1911.06189 [hep-ph], pp. 1-10. 2020. ISSN 1824-8039. [url]@

author = {A. Behring and J. Blümlein and A. De Freitas and A. Goedicke and S. Klein and A. van Manteuffel and C. Schneider and K. Schönwald},

title = {{The Polarized Three-Loop Anomalous Dimensions from a Massive Calculation}},

booktitle = {{14th International Symposium on Radiative Corrections (RADCOR2019)}},

language = {english},

series = {POS(RADCOR2019)047},

number = {arXiv:1911.06189 [hep-ph]},

pages = {1--10},

isbn_issn = {ISSN 1824-8039},

year = {2020},

editor = {D. Kosower and M. Cacciari},

refereed = {yes},

length = {10},

url = {https://doi.org/10.22323/1.375.0047}

}

**inproceedings**{RISC6016,author = {A. Behring and J. Blümlein and A. De Freitas and A. Goedicke and S. Klein and A. van Manteuffel and C. Schneider and K. Schönwald},

title = {{The Polarized Three-Loop Anomalous Dimensions from a Massive Calculation}},

booktitle = {{14th International Symposium on Radiative Corrections (RADCOR2019)}},

language = {english},

series = {POS(RADCOR2019)047},

number = {arXiv:1911.06189 [hep-ph]},

pages = {1--10},

isbn_issn = {ISSN 1824-8039},

year = {2020},

editor = {D. Kosower and M. Cacciari},

refereed = {yes},

length = {10},

url = {https://doi.org/10.22323/1.375.0047}

}

[Schneider]

### The three-loop polarized pure singlet operator matrix element with two different masses

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

Nuclear Physics B 952(114916), pp. 1-18. 2020. ISSN 0550-3213. arXiv:1911.11630 [hep-ph]. [url]@

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

title = {{The three-loop polarized pure singlet operator matrix element with two different masses}},

language = {english},

journal = {Nuclear Physics B },

volume = {952},

number = {114916},

pages = {1--18},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:1911.11630 [hep-ph]},

refereed = {yes},

length = {18},

url = {https://doi.org/10.1016/j.nuclphysb.2020.114916}

}

**article**{RISC6017,author = {J. Ablinger and J. Blümlein and A. De Freitas and M. Saragnese and C. Schneider and K. Schönwald},

title = {{The three-loop polarized pure singlet operator matrix element with two different masses}},

language = {english},

journal = {Nuclear Physics B },

volume = {952},

number = {114916},

pages = {1--18},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:1911.11630 [hep-ph]},

refereed = {yes},

length = {18},

url = {https://doi.org/10.1016/j.nuclphysb.2020.114916}

}

[Schneider]

### 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. ?-?. 2020. arXiv:1906.05829 [hep-ph]. [url]@

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 = {2020},

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

editor = {?},

refereed = {no},

length = {8},

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

}

**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 = {2020},

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

editor = {?},

refereed = {no},

length = {8},

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

}

[Schneider]

### The three-loop single mass polarized pure singlet operator matrix element

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

Nuclear Physics B 953(114945), pp. 1-25. 2020. ISSN 0550-3213. arXiv:1912.02536 [hep-ph]. [url]@

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

title = {{The three-loop single mass polarized pure singlet operator matrix element}},

language = {english},

journal = {Nuclear Physics B},

volume = {953},

number = {114945},

pages = {1--25},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:1912.02536 [hep-ph]},

refereed = {yes},

length = {25},

url = {https://doi.org/10.1016/j.nuclphysb.2020.114945}

}

**article**{RISC6075,author = {J. Ablinger and A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schönwald},

title = {{The three-loop single mass polarized pure singlet operator matrix element}},

language = {english},

journal = {Nuclear Physics B},

volume = {953},

number = {114945},

pages = {1--25},

isbn_issn = {ISSN 0550-3213},

year = {2020},

note = {arXiv:1912.02536 [hep-ph]},

refereed = {yes},

length = {25},

url = {https://doi.org/10.1016/j.nuclphysb.2020.114945}

}

[Smoot]

### A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5

#### Nicolas Allen Smoot

Submitted, pp. 1-32. 2020. Submitted. [pdf]@

author = {Nicolas Allen Smoot},

title = {{A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5}},

language = {english},

abstract = {In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $\omega(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.},

journal = {Submitted},

pages = {1--32},

isbn_issn = {Submitted},

year = {2020},

refereed = {yes},

length = {32}

}

**article**{RISC6079,author = {Nicolas Allen Smoot},

title = {{A Single-Variable Proof of the Omega SPT Congruence Family Over Powers of 5}},

language = {english},

abstract = {In 2018 Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third order mock theta function $\omega(q)$. Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 $\mathbb{Z}[X]$-module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of $\mathbb{Z}[X]$. To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic growth. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.},

journal = {Submitted},

pages = {1--32},

isbn_issn = {Submitted},

year = {2020},

refereed = {yes},

length = {32}

}

### 2019

[Ablinger]

### Discovering and Proving Infinite Pochhammer Sum Identities

#### J. Ablinger

Experimental Mathematics, pp. 1-15. 2019. Taylor & Francis, 10.1080/10586458.2019.1627254. [url]@

author = {J. Ablinger},

title = {{Discovering and Proving Infinite Pochhammer Sum Identities}},

language = {english},

journal = {Experimental Mathematics},

pages = {1--15},

publisher = {Taylor & Francis},

isbn_issn = {?},

year = {2019},

note = {10.1080/10586458.2019.1627254},

refereed = {yes},

length = {15},

url = {https://doi.org/10.1080/10586458.2019.1627254}

}

**article**{RISC5896,author = {J. Ablinger},

title = {{Discovering and Proving Infinite Pochhammer Sum Identities}},

language = {english},

journal = {Experimental Mathematics},

pages = {1--15},

publisher = {Taylor & Francis},

isbn_issn = {?},

year = {2019},

note = {10.1080/10586458.2019.1627254},

refereed = {yes},

length = {15},

url = {https://doi.org/10.1080/10586458.2019.1627254}

}

[Ablinger]

### Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.

#### J. Ablinger

arXiv. Technical report, 2019. [url]@

author = {J. Ablinger},

title = {{Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.}},

language = {english},

year = {2019},

institution = {arXiv},

length = {5},

url = {https://arxiv.org/pdf/1908.06631.pdf}

}

**techreport**{RISC5968,author = {J. Ablinger},

title = {{Proving two conjectural series for $\zeta(7)$ and discovering more series for $\zeta(7)$.}},

language = {english},

year = {2019},

institution = {arXiv},

length = {5},

url = {https://arxiv.org/pdf/1908.06631.pdf}

}

[Berkovich]

### Polynomial Identities Implying Capparelli's Partition Theorems

#### Ali Kemal Uncu, Alexander Berkovich

Accepted - Journal of Number Theory, pp. -. 2019. N/A. [url]@

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Polynomial Identities Implying Capparelli's Partition Theorems }},

language = {english},

journal = {Accepted - Journal of Number Theory},

pages = {--},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {21},

url = {https://arxiv.org/pdf/1807.10974.pdf}

}

**article**{RISC5790,author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Polynomial Identities Implying Capparelli's Partition Theorems }},

language = {english},

journal = {Accepted - Journal of Number Theory},

pages = {--},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {21},

url = {https://arxiv.org/pdf/1807.10974.pdf}

}

[Berkovich]

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

#### Ali Kemal Uncu, Alexander Berkovich

ArXiv e-prints (accepted), pp. 1-10. 2019. N/A. [url]@

author = {Ali Kemal Uncu and Alexander Berkovich},

title = {{Refined q-Trinomial Coefficients and Two Infinite Hierarchies 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 hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

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

pages = {1--10},

isbn_issn = {N/A},

year = {2019},

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 Hierarchies 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 hierarchies contains an identity which is equivalent to Capparelli's first Partition Theorem. },

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

pages = {1--10},

isbn_issn = {N/A},

year = {2019},

refereed = {yes},

length = {10},

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

}