# ARC: Automated Reasoning in The Class [Erasmus+ Project]

### Project Lead

### Project Duration

01/10/2019 - 31/08/2022## Partners

### EU

## Publications

### 2024

[Schneider]

### Representation of hypergeometric products of higher nesting depths in difference rings

#### E.D. Ocansey, C. Schneider

J. Symb. Comput. 120, pp. 1-50. 2024. ISSN: 0747-7171. arXiv:2011.08775 [cs.SC]. [doi]@

author = {E.D. Ocansey and C. Schneider},

title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},

language = {english},

journal = {J. Symb. Comput.},

volume = {120},

pages = {1--50},

isbn_issn = {ISSN: 0747-7171},

year = {2024},

note = {arXiv:2011.08775 [cs.SC]},

refereed = {yes},

length = {50},

url = {https://doi.org/10.1016/j.jsc.2023.03.002}

}

**article**{RISC6688,author = {E.D. Ocansey and C. Schneider},

title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},

language = {english},

journal = {J. Symb. Comput.},

volume = {120},

pages = {1--50},

isbn_issn = {ISSN: 0747-7171},

year = {2024},

note = {arXiv:2011.08775 [cs.SC]},

refereed = {yes},

length = {50},

url = {https://doi.org/10.1016/j.jsc.2023.03.002}

}

### 2023

[Banerjee]

### Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach

#### Koustav Banerjee

Enumerative Combinatorics and Applications 3, pp. 1-4. 2023. ISSN 2710-2335. [doi]@

author = {Koustav Banerjee},

title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},

language = {english},

journal = {Enumerative Combinatorics and Applications},

volume = {3},

pages = {1--4},

isbn_issn = {ISSN 2710-2335},

year = {2023},

refereed = {yes},

length = {5},

url = {https://doi.org/10.54550/ECA2023V3S2R12}

}

**article**{RISC6701,author = {Koustav Banerjee},

title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},

language = {english},

journal = {Enumerative Combinatorics and Applications},

volume = {3},

pages = {1--4},

isbn_issn = {ISSN 2710-2335},

year = {2023},

refereed = {yes},

length = {5},

url = {https://doi.org/10.54550/ECA2023V3S2R12}

}

[Banerjee]

### Inequalities for the modified Bessel function of first kind of non-negative order

#### K. Banerjee

Journal of Mathematical Analysis and Applications 524, pp. 1-28. 2023. Elsevier, ISSN 1096-0813. [doi]@

author = {K. Banerjee},

title = {{Inequalities for the modified Bessel function of first kind of non-negative order}},

language = {english},

journal = {Journal of Mathematical Analysis and Applications},

volume = {524},

pages = {1--28},

publisher = {Elsevier},

isbn_issn = {ISSN 1096-0813},

year = {2023},

refereed = {yes},

length = {28},

url = {https://doi.org/10.1016/j.jmaa.2023.127082}

}

**article**{RISC6700,author = {K. Banerjee},

title = {{Inequalities for the modified Bessel function of first kind of non-negative order}},

language = {english},

journal = {Journal of Mathematical Analysis and Applications},

volume = {524},

pages = {1--28},

publisher = {Elsevier},

isbn_issn = {ISSN 1096-0813},

year = {2023},

refereed = {yes},

length = {28},

url = {https://doi.org/10.1016/j.jmaa.2023.127082}

}

[Banerjee]

### 2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family

#### K. Banerjee, N.A. Smoot

Technical report no. 23-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {K. Banerjee and N.A. Smoot},

title = {{2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family}},

language = {english},

abstract = {Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and present a second congruence family by powers of 7 which we conjecture, and which may be amenable to similar techniques.},

number = {23-10},

year = {2023},

month = {August},

keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},

length = {35},

license = {CC BY 4.0 International},

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**{RISC6713,author = {K. Banerjee and N.A. Smoot},

title = {{2-Elongated Plane Partitions and Powers of 7: The Localization Method Applied to a Genus 1 Congruence Family}},

language = {english},

abstract = {Over the last century, a large variety of infinite congruence families have been discovered and studied, exhibiting a great variety with respect to their difficulty. Major complicating factors arise from the topology of the associated modular curve: classical techniques are sufficient when the associated curve has cusp count 2 and genus 0. Recent work has led to new techniques that have proven useful when the associated curve has cusp count greater than 2 and genus 0. We show here that these techniques may be adapted in the case of positive genus. In particular, we examine a congruence family over the 2-elongated plane partition diamond counting function $d_2(n)$ by powers of 7, for which the associated modular curve has cusp count 4 and genus 1. We compare our method with other techniques for proving genus 1 congruence families, and present a second congruence family by powers of 7 which we conjecture, and which may be amenable to similar techniques.},

number = {23-10},

year = {2023},

month = {August},

keywords = {Partition congruences, infinite congruence family, modular functions, plane partitions, partition analysis, modular curve, Riemann surface},

length = {35},

license = {CC BY 4.0 International},

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)}

}

[Buchberger]

### Is ChatGPT Smarter Than Master’s Applicants?

#### Bruno Buchberger

Technical report no. 23-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Bruno Buchberger},

title = {{Is ChatGPT Smarter Than Master’s Applicants?}},

language = {English},

abstract = {During the selection procedure for a particular informatics fellowship program sponsored by Upper Austrian companies, I ask the applicants a couple of simple technical questions about programming, etc., in a Zoom meeting. I put the same questions to the dialogue system ChatGPT, [ChatGPT]. The result surprised me: Nearly all answers of ChatGPT were totally correct and nicely explained. Also, in the dialogues to clarify some critical points in the answers, the explanations by ChatGPT were amazingly clear and goal-oriented.In comparison: I tried out the same questions in the personal Zoom interviews with approximately 30 applicants from five countries. Only the top three candidates (with a GPA of 1.0, i.e., the highest possible GPA in their bachelor’s study) performed approximately equally well in the interview. All the others performed (far) worse than ChatGPT. And, of course, all answers from ChatGPT came within 1 to 10 seconds, whereas most of the human applicants' answers needed lengthy and arduous dialogues.I am particularly impressed by the ability of ChatGPT to extract meaningful and well-structured programs from problem specifications in natural language. In this experiment, I also added some questions that ask for proofs for simple statements in natural language, which I do not ask in the student's interviews. The performance of ChatGPT was quite impressive as far as formalization and propositional logic are concerned. In examples where predicate logic reasoning is necessary, the ChatGPT answers are not (yet?) perfect. I am pleased to see that ChatGPT tries to present the proofs in a “natural style” This is something that I had as one of my main goals when I initiated the Theorema project in 1995. I think we already achieved this in the early stage of Theorema, and we performed this slightly better and more systematically than ChatGPT does.I also tried to develop a natural language input facility for Theorema in 2017, i.e., a tool to formalize natural language statements in predicate logic. However, I could not continue this research for a couple of reasons. Now I see that ChatGPT achieved this goal. Thus, I think that the following combination of methods could result in a significant leap forward:- the “natural style” proving methods that we developed within Theorema (for the automated generation of programs from specifications, the automated verification of programs in the frame of knowledge, and the automated proof of theorems in theories), in particular, my “Lazy Thinking Method” for algorithm synthesis from specifications- and the natural language formalization techniques of ChatGPT.I propose this as a research project topic and invite colleagues and students to contact me and join me in this effort: Buchberger.bruno@gmail.com.},

number = {23-04},

year = {2023},

month = {January},

keywords = {ChatGPT, automated programming, program synthesis, automated proving, formalization of natural language, master's screening},

length = {30},

license = {CC BY 4.0 International},

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**{RISC6684,author = {Bruno Buchberger},

title = {{Is ChatGPT Smarter Than Master’s Applicants?}},

language = {English},

abstract = {During the selection procedure for a particular informatics fellowship program sponsored by Upper Austrian companies, I ask the applicants a couple of simple technical questions about programming, etc., in a Zoom meeting. I put the same questions to the dialogue system ChatGPT, [ChatGPT]. The result surprised me: Nearly all answers of ChatGPT were totally correct and nicely explained. Also, in the dialogues to clarify some critical points in the answers, the explanations by ChatGPT were amazingly clear and goal-oriented.In comparison: I tried out the same questions in the personal Zoom interviews with approximately 30 applicants from five countries. Only the top three candidates (with a GPA of 1.0, i.e., the highest possible GPA in their bachelor’s study) performed approximately equally well in the interview. All the others performed (far) worse than ChatGPT. And, of course, all answers from ChatGPT came within 1 to 10 seconds, whereas most of the human applicants' answers needed lengthy and arduous dialogues.I am particularly impressed by the ability of ChatGPT to extract meaningful and well-structured programs from problem specifications in natural language. In this experiment, I also added some questions that ask for proofs for simple statements in natural language, which I do not ask in the student's interviews. The performance of ChatGPT was quite impressive as far as formalization and propositional logic are concerned. In examples where predicate logic reasoning is necessary, the ChatGPT answers are not (yet?) perfect. I am pleased to see that ChatGPT tries to present the proofs in a “natural style” This is something that I had as one of my main goals when I initiated the Theorema project in 1995. I think we already achieved this in the early stage of Theorema, and we performed this slightly better and more systematically than ChatGPT does.I also tried to develop a natural language input facility for Theorema in 2017, i.e., a tool to formalize natural language statements in predicate logic. However, I could not continue this research for a couple of reasons. Now I see that ChatGPT achieved this goal. Thus, I think that the following combination of methods could result in a significant leap forward:- the “natural style” proving methods that we developed within Theorema (for the automated generation of programs from specifications, the automated verification of programs in the frame of knowledge, and the automated proof of theorems in theories), in particular, my “Lazy Thinking Method” for algorithm synthesis from specifications- and the natural language formalization techniques of ChatGPT.I propose this as a research project topic and invite colleagues and students to contact me and join me in this effort: Buchberger.bruno@gmail.com.},

number = {23-04},

year = {2023},

month = {January},

keywords = {ChatGPT, automated programming, program synthesis, automated proving, formalization of natural language, master's screening},

length = {30},

license = {CC BY 4.0 International},

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)}

}

[Cerna]

### Anti-unification and Generalization: a Survey

#### David Cerna, Temur Kutsia

In: Proceedings of IJCAI 2023 - 32nd International Joint Conference on Artifical Intelligence, Edith Elkind (ed.), pp. 6563-6573. 2023. ijcai.org, ISBN 978-1-956792-03-4 . [doi]@

author = {David Cerna and Temur Kutsia},

title = {{Anti-unification and Generalization: a Survey}},

booktitle = {{Proceedings of IJCAI 2023 - 32nd International Joint Conference on Artifical Intelligence}},

language = {english},

pages = {6563--6573},

publisher = {ijcai.org},

isbn_issn = {ISBN 978-1-956792-03-4 },

year = {2023},

editor = {Edith Elkind},

refereed = {yes},

length = {11},

url = {https://doi.org/10.24963/ijcai.2023/736}

}

**inproceedings**{RISC6743,author = {David Cerna and Temur Kutsia},

title = {{Anti-unification and Generalization: a Survey}},

booktitle = {{Proceedings of IJCAI 2023 - 32nd International Joint Conference on Artifical Intelligence}},

language = {english},

pages = {6563--6573},

publisher = {ijcai.org},

isbn_issn = {ISBN 978-1-956792-03-4 },

year = {2023},

editor = {Edith Elkind},

refereed = {yes},

length = {11},

url = {https://doi.org/10.24963/ijcai.2023/736}

}

[de Freitas]

### Analytic results on the massive three-loop form factors: quarkonic contributions

#### J. Bluemlein, A. De Freitas, P. Marquard, N. Rana, C. Schneider

Physical Review D to appear, pp. ?-?. July 2023. ISSN 2470-0029. arXiv:2307.02983 [hep-ph]. [doi]@

author = {J. Bluemlein and A. De Freitas and P. Marquard and N. Rana and C. Schneider},

title = {{Analytic results on the massive three-loop form factors: quarkonic contributions}},

language = {english},

abstract = {The quarkonic contributions to the three--loop heavy-quark form factors for vector, axial-vector, scalar and pseudoscalar currents are described by closed form difference equations for the expansion coefficients in the limit of small virtualities $q^2/m^2$. A part of the contributions can be solved analytically and expressed in terms of harmonic and cyclotomic harmonic polylogarithms and square-root valued iterated integrals. Other contributions obey equations which are not first--order factorizable. For them still infinite series expansions around the singularities of the form factors can be obtained by matching the expansions at intermediate points and using differential equations which are obeyed directly by the form factors and are derived by guessing algorithms. One may determine all expansion coefficients for $q^2/m^2 rightarrow infty$ analytically in terms of multiple zeta values. By expanding around the threshold and pseudo--threshold, the corresponding constants are multiple zeta values supplemented by a finite amount of new constants, which can be computed at high precision. For a part of these coefficients, the infinite series in front of these constants may be even resummed into harmonic polylogarithms. In this way, one obtains a deeper analytic description of the massive form factors, beyond their pure numerical evaluation. The calculations of these analytic results are based on sophisticated computer algebra techniques. We also compare our results with numerical results in the literature.},

journal = {Physical Review D},

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN 2470-0029},

year = {2023},

month = {July},

note = {arXiv:2307.02983 [hep-ph]},

refereed = {yes},

keywords = {form factor, Feynman diagram, computer algebra, holonomic properties, difference equations, differential equations, symbolic summation, numerical matching, analytic continuation, guessing, PSLQ},

length = {92},

url = {https://doi.org/10.35011/risc.23-08}

}

**article**{RISC6742,author = {J. Bluemlein and A. De Freitas and P. Marquard and N. Rana and C. Schneider},

title = {{Analytic results on the massive three-loop form factors: quarkonic contributions}},

language = {english},

abstract = {The quarkonic contributions to the three--loop heavy-quark form factors for vector, axial-vector, scalar and pseudoscalar currents are described by closed form difference equations for the expansion coefficients in the limit of small virtualities $q^2/m^2$. A part of the contributions can be solved analytically and expressed in terms of harmonic and cyclotomic harmonic polylogarithms and square-root valued iterated integrals. Other contributions obey equations which are not first--order factorizable. For them still infinite series expansions around the singularities of the form factors can be obtained by matching the expansions at intermediate points and using differential equations which are obeyed directly by the form factors and are derived by guessing algorithms. One may determine all expansion coefficients for $q^2/m^2 rightarrow infty$ analytically in terms of multiple zeta values. By expanding around the threshold and pseudo--threshold, the corresponding constants are multiple zeta values supplemented by a finite amount of new constants, which can be computed at high precision. For a part of these coefficients, the infinite series in front of these constants may be even resummed into harmonic polylogarithms. In this way, one obtains a deeper analytic description of the massive form factors, beyond their pure numerical evaluation. The calculations of these analytic results are based on sophisticated computer algebra techniques. We also compare our results with numerical results in the literature.},

journal = {Physical Review D},

volume = {to appear},

pages = {?--?},

isbn_issn = {ISSN 2470-0029},

year = {2023},

month = {July},

note = {arXiv:2307.02983 [hep-ph]},

refereed = {yes},

keywords = {form factor, Feynman diagram, computer algebra, holonomic properties, difference equations, differential equations, symbolic summation, numerical matching, analytic continuation, guessing, PSLQ},

length = {92},

url = {https://doi.org/10.35011/risc.23-08}

}

[de Freitas]

### Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering

#### J. Ablinger, A. Behring, J. Bluemlein, A. De Freitas, A. Goedicke, A. von Manteuffel, C. Schneider, K. Schoenwald

In: Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology , Giulio Falcioni (ed.), PoS RADCOR2023046, pp. 1-7. June 2023. ISSN 1824-8039. arXiv:2306.16550 [hep-ph]. [doi]@

author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering}},

booktitle = {{Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology }},

language = {english},

abstract = {We report on recent progress in calculating the three loop QCD corrections of the heavy flavor contributions in deep--inelastic scattering and the massive operator matrix elements of the variable flavor number scheme. Notably we deal with the operator matrix elements $A_{gg,Q}^{(3)}$ and $A_{Qg}^{(3)}$ and technical steps to their calculation. In particular, a new method to obtain the inverse Mellin transform without computing the corresponding $N$--space expressions is discussed.},

series = {PoS},

volume = {RADCOR2023},

number = {046},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2023},

month = {June},

note = {arXiv:2306.16550 [hep-ph]},

editor = {Giulio Falcioni},

refereed = {no},

keywords = {deep-inelastic scattering, 3-loop Feynman diagrams, (inverse) Mellin transform, binomial sums},

length = {7},

url = { https://doi.org/10.22323/1.432.0046 }

}

**inproceedings**{RISC6748,author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. Goedicke and A. von Manteuffel and C. Schneider and K. Schoenwald},

title = {{Recent 3-Loop Heavy Flavor Corrections to Deep-Inelastic Scattering}},

booktitle = {{Proc. of the 16th International Symposium on Radiative Corrections: Applications of Quantum Field Theory to Phenomenology }},

language = {english},

abstract = {We report on recent progress in calculating the three loop QCD corrections of the heavy flavor contributions in deep--inelastic scattering and the massive operator matrix elements of the variable flavor number scheme. Notably we deal with the operator matrix elements $A_{gg,Q}^{(3)}$ and $A_{Qg}^{(3)}$ and technical steps to their calculation. In particular, a new method to obtain the inverse Mellin transform without computing the corresponding $N$--space expressions is discussed.},

series = {PoS},

volume = {RADCOR2023},

number = {046},

pages = {1--7},

isbn_issn = {ISSN 1824-8039},

year = {2023},

month = {June},

note = {arXiv:2306.16550 [hep-ph]},

editor = {Giulio Falcioni},

refereed = {no},

keywords = {deep-inelastic scattering, 3-loop Feynman diagrams, (inverse) Mellin transform, binomial sums},

length = {7},

url = { https://doi.org/10.22323/1.432.0046 }

}

[Dominici]

### Linear functionals and $Delta$- coherent pairs of the second kind

#### Diego Dominici and Francisco Marcellan

Technical report no. 23-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Diego Dominici and Francisco Marcellan},

title = {{Linear functionals and $Delta$- coherent pairs of the second kind}},

language = {english},

abstract = {We classify all the emph{$Delta$-}coherent pairs of measures of the secondkind on the real line. We obtain $5$ cases, corresponding to all the familiesof discrete semiclassical orthogonal polynomials of class $sleq1.$},

number = {23-02},

year = {2023},

month = {February},

keywords = { Discrete orthogonal polynomials, discrete semiclassical functionals, discrete Sobolev inner products, coherent pairs of discrete measures, coherent pairs of second kind for discrete measures.},

length = {24},

license = {CC BY 4.0 International},

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**{RISC6677,author = {Diego Dominici and Francisco Marcellan},

title = {{Linear functionals and $Delta$- coherent pairs of the second kind}},

language = {english},

abstract = {We classify all the emph{$Delta$-}coherent pairs of measures of the secondkind on the real line. We obtain $5$ cases, corresponding to all the familiesof discrete semiclassical orthogonal polynomials of class $sleq1.$},

number = {23-02},

year = {2023},

month = {February},

keywords = { Discrete orthogonal polynomials, discrete semiclassical functionals, discrete Sobolev inner products, coherent pairs of discrete measures, coherent pairs of second kind for discrete measures.},

length = {24},

license = {CC BY 4.0 International},

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)}

}

[Dominici]

### Recurrence relations for the moments of discrete semiclassical functionals of class $sleq2.$

#### Diego Dominici

Technical report no. 23-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Diego Dominici },

title = {{Recurrence relations for the moments of discrete semiclassical functionals of class $sleq2.$}},

language = {english},

abstract = {We study recurrence relations satisfied by the moments $lambda_{n}left(zright) $ of discrete linear functionals whose first moment satisfies aholonomic differential equation. We consider all cases when the order of theODE is less or equal than $3$.},

number = {23-05},

year = {2023},

month = {March},

keywords = {Discrete orthogonal polynomials, discrete semiclassical functionals, moments.},

length = {81},

license = {CC BY 4.0 International},

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**{RISC6687,author = {Diego Dominici },

title = {{Recurrence relations for the moments of discrete semiclassical functionals of class $sleq2.$}},

language = {english},

abstract = {We study recurrence relations satisfied by the moments $lambda_{n}left(zright) $ of discrete linear functionals whose first moment satisfies aholonomic differential equation. We consider all cases when the order of theODE is less or equal than $3$.},

number = {23-05},

year = {2023},

month = {March},

keywords = {Discrete orthogonal polynomials, discrete semiclassical functionals, moments.},

length = {81},

license = {CC BY 4.0 International},

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)}

}

[Jimenez Pastor]

### An extension of holonomic sequences: $C^2$-finite sequences

#### A. Jimenez-Pastor, P. Nuspl, V. Pillwein

Journal of Symbolic Computation 116, pp. 400-424. 2023. ISSN: 0747-7171.@

author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},

title = {{An extension of holonomic sequences: $C^2$-finite sequences}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {116},

pages = {400--424},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

refereed = {yes},

length = {25}

}

**article**{RISC6636,author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},

title = {{An extension of holonomic sequences: $C^2$-finite sequences}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {116},

pages = {400--424},

isbn_issn = {ISSN: 0747-7171},

year = {2023},

refereed = {yes},

length = {25}

}

[Kauers]

### Order bounds for $C^2$-finite sequences

#### M. Kauers, P. Nuspl, V. Pillwein

Technical report no. 23-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {M. Kauers and P. Nuspl and V. Pillwein},

title = {{Order bounds for $C^2$-finite sequences}},

language = {english},

abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},

number = {23-03},

year = {2023},

month = {February},

keywords = {Difference equations, holonomic sequences, closure properties, algorithms},

length = {16},

license = {CC BY 4.0 International},

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**{RISC6683,author = {M. Kauers and P. Nuspl and V. Pillwein},

title = {{Order bounds for $C^2$-finite sequences}},

language = {english},

abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},

number = {23-03},

year = {2023},

month = {February},

keywords = {Difference equations, holonomic sequences, closure properties, algorithms},

length = {16},

license = {CC BY 4.0 International},

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)}

}

[Kutsia]

### Nominal AC-Matching

#### Mauricio Ayala-Rincón, Maribel Fernández, Gabriel Ferreira Silva, Temur Kutsia, and Daniele Nantes-Sobrinho

In: Proceedings of the 16th International Conference on Intelligent Computer Mathematics, CICM 2023, Catherine Dubois and Manfred Kerber (ed.), Lecture Notes in Aritificial Intelligence 14101, pp. 53-68. 2023. Springer, ISBN 978-3-031-42752-7. [doi]@

author = {Mauricio Ayala-Rincón and Maribel Fernández and Gabriel Ferreira Silva and Temur Kutsia and and Daniele Nantes-Sobrinho},

title = {{Nominal AC-Matching}},

booktitle = {{Proceedings of the 16th International Conference on Intelligent Computer Mathematics, CICM 2023}},

language = {english},

series = {Lecture Notes in Aritificial Intelligence},

volume = {14101},

pages = {53--68},

publisher = {Springer},

isbn_issn = {ISBN 978-3-031-42752-7},

year = {2023},

editor = {Catherine Dubois and Manfred Kerber},

refereed = {yes},

length = {16},

url = {https://doi.org/10.1007/978-3-031-42753-4_4}

}

**inproceedings**{RISC6744,author = {Mauricio Ayala-Rincón and Maribel Fernández and Gabriel Ferreira Silva and Temur Kutsia and and Daniele Nantes-Sobrinho},

title = {{Nominal AC-Matching}},

booktitle = {{Proceedings of the 16th International Conference on Intelligent Computer Mathematics, CICM 2023}},

language = {english},

series = {Lecture Notes in Aritificial Intelligence},

volume = {14101},

pages = {53--68},

publisher = {Springer},

isbn_issn = {ISBN 978-3-031-42752-7},

year = {2023},

editor = {Catherine Dubois and Manfred Kerber},

refereed = {yes},

length = {16},

url = {https://doi.org/10.1007/978-3-031-42753-4_4}

}

[Mitteramskogler]

### The algebro-geometric method: Solving algebraic differential equations by parametrizations

#### S. Falkensteiner, J.J. Mitteramskogler, R. Sendra, F. Winkler

Bulletin of the American Mathematical Society, pp. 1-41. 2023. ISSN 0273-0979.@

author = {S. Falkensteiner and J.J. Mitteramskogler and R. Sendra and F. Winkler},

title = {{The algebro-geometric method: Solving algebraic differential equations by parametrizations}},

language = {english},

journal = {Bulletin of the American Mathematical Society},

pages = {1--41},

isbn_issn = {ISSN 0273-0979},

year = {2023},

refereed = {yes},

length = {41}

}

**article**{RISC6507,author = {S. Falkensteiner and J.J. Mitteramskogler and R. Sendra and F. Winkler},

title = {{The algebro-geometric method: Solving algebraic differential equations by parametrizations}},

language = {english},

journal = {Bulletin of the American Mathematical Society},

pages = {1--41},

isbn_issn = {ISSN 0273-0979},

year = {2023},

refereed = {yes},

length = {41}

}

[Mitteramskogler]

### General solutions of first-order algebraic ODEs in simple constant extensions

#### J. J. Mitteramskogler, F. Winkler

Journal of Systems Science and Complexity (JSSC), pp. 0-0. 2023. 1009-6124.@

author = {J. J. Mitteramskogler and F. Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

journal = {Journal of Systems Science and Complexity (JSSC)},

pages = {0--0},

isbn_issn = {1009-6124},

year = {2023},

refereed = {yes},

length = {0}

}

**article**{RISC6674,author = {J. J. Mitteramskogler and F. Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

journal = {Journal of Systems Science and Complexity (JSSC)},

pages = {0--0},

isbn_issn = {1009-6124},

year = {2023},

refereed = {yes},

length = {0}

}

[Nuspl]

### Algorithms for linear recurrence sequences

#### P. Nuspl

Johannes Kepler University Linz. PhD Thesis. 2023. [pdf]@

author = {P. Nuspl},

title = {{Algorithms for linear recurrence sequences}},

language = {english},

year = {2023},

translation = {0},

school = {Johannes Kepler University Linz},

length = {129}

}

**phdthesis**{RISC6711,author = {P. Nuspl},

title = {{Algorithms for linear recurrence sequences}},

language = {english},

year = {2023},

translation = {0},

school = {Johannes Kepler University Linz},

length = {129}

}

[Paule]

### Ramanujan and Computer Algebra

#### Peter Paule

In: Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, K. Alladi, G.E. Andrews, B. Berndt, F. Garvan, K. Ono, P. Paule, S. Ole Warnaar, Ae Ja Yee (ed.), pp. -. 2023. Springer, ISBN x. [pdf]@

author = {Peter Paule},

title = {{Ramanujan and Computer Algebra}},

booktitle = {{Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence}},

language = {english},

pages = {--},

publisher = {Springer},

isbn_issn = {ISBN x},

year = {2023},

editor = {K. Alladi and G.E. Andrews and B. Berndt and F. Garvan and K. Ono and P. Paule and S. Ole Warnaar and Ae Ja Yee },

refereed = {yes},

length = {0}

}

**incollection**{RISC6678,author = {Peter Paule},

title = {{Ramanujan and Computer Algebra}},

booktitle = {{Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence}},

language = {english},

pages = {--},

publisher = {Springer},

isbn_issn = {ISBN x},

year = {2023},

editor = {K. Alladi and G.E. Andrews and B. Berndt and F. Garvan and K. Ono and P. Paule and S. Ole Warnaar and Ae Ja Yee },

refereed = {yes},

length = {0}

}

[Paule]

### Interview with Peter Paule

#### Toufik Mansour and Peter Paule

Enumerative Combinatorics and Applications ECA 3:1(#S3I1), pp. -. 2023. ISSN 2710-2335. [doi]@

author = {Toufik Mansour and Peter Paule},

title = {{Interview with Peter Paule}},

language = {english},

journal = {Enumerative Combinatorics and Applications },

volume = {ECA 3:1},

number = {#S3I1},

pages = {--},

isbn_issn = {ISSN 2710-2335},

year = {2023},

refereed = {yes},

length = {0},

url = {http://doi.org/10.54550/ECA2023V3S1I1}

}

**article**{RISC6679,author = {Toufik Mansour and Peter Paule},

title = {{Interview with Peter Paule}},

language = {english},

journal = {Enumerative Combinatorics and Applications },

volume = {ECA 3:1},

number = {#S3I1},

pages = {--},

isbn_issn = {ISSN 2710-2335},

year = {2023},

refereed = {yes},

length = {0},

url = {http://doi.org/10.54550/ECA2023V3S1I1}

}

[Schneider]

### Hypergeometric Structures in Feynman Integrals

#### J. Blümlein, C. Schneider, M. Saragnese

Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science" in press, pp. ?-?. 2023. ISSN 1573-7470. arXiv:2111.15501 [math-ph]. [doi]@

author = {J. Blümlein and C. Schneider and M. Saragnese},

title = {{Hypergeometric Structures in Feynman Integrals}},

language = {english},

abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},

journal = {Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science"},

volume = {in press},

pages = {?--?},

isbn_issn = {ISSN 1573-7470},

year = {2023},

note = {arXiv:2111.15501 [math-ph]},

refereed = {yes},

keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},

length = {55},

url = {https://doi.org/10.1007/s10472-023-09831-8}

}

**article**{RISC6643,author = {J. Blümlein and C. Schneider and M. Saragnese},

title = {{Hypergeometric Structures in Feynman Integrals}},

language = {english},

abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},

journal = {Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science"},

volume = {in press},

pages = {?--?},

isbn_issn = {ISSN 1573-7470},

year = {2023},

note = {arXiv:2111.15501 [math-ph]},

refereed = {yes},

keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},

length = {55},

url = {https://doi.org/10.1007/s10472-023-09831-8}

}

[Schneider]

### Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators

#### C. Schneider

In: ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation, Gabriela Jeronimo (ed.), pp. 498-507. July 2023. ACM, ISBN 9798400700392. arXiv:2302.03563 [cs.SC]. [doi]@

author = {C. Schneider},

title = {{Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators}},

booktitle = {{ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $RPiSigma$-ring extensions that are built over general $PiSigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},

pages = {498--507},

publisher = {ACM},

isbn_issn = {ISBN 9798400700392},

year = {2023},

month = {July},

note = {arXiv:2302.03563 [cs.SC]},

editor = {Gabriela Jeronimo},

refereed = {yes},

keywords = {telescoping, difference rings, reduced denominators, nested sums},

length = {10},

url = {https://doi.org/10.1145/3597066.3597073}

}

**inproceedings**{RISC6699,author = {C. Schneider},

title = {{Refined telescoping algorithms in $RPiSigma$-extensions to reduce the degrees of the denominators}},

booktitle = {{ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $RPiSigma$-ring extensions that are built over general $PiSigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},

pages = {498--507},

publisher = {ACM},

isbn_issn = {ISBN 9798400700392},

year = {2023},

month = {July},

note = {arXiv:2302.03563 [cs.SC]},

editor = {Gabriela Jeronimo},

refereed = {yes},

keywords = {telescoping, difference rings, reduced denominators, nested sums},

length = {10},

url = {https://doi.org/10.1145/3597066.3597073}

}