RISC Reports Series

2024

[Schneider]

Creative Telescoping for Hypergeometric Double Sums

P. Paule, C. Schneider

Technical report no. 24-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2024. arXiv:2401.16314 [cs.SC]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6894,
author = {P. Paule and C. Schneider},
title = {{Creative Telescoping for Hypergeometric Double Sums}},
language = {english},
abstract = {We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which guarantees the applicability of our method for many input sums. In addition, we elaborate new techniques to optimize the underlying key task of our method to compute rational solutions of parameterized linear recurrences.},
number = {24-01},
year = {2024},
month = {January},
note = {arXiv:2401.16314 [cs.SC]},
keywords = {creative telescoping; symbolic summation, hypergeometric multi-sums, contiguous relations, parameterized recurrences, rational solutions},
length = {26},
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)}
}
[de Freitas]

The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$

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

Technical report no. 24-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2024. arXiv:2403.00513 [[hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC7042,
author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},
language = {english},
abstract = {The non-first-order-factorizable contributions to the unpolarized and polarized massive operator matrix elements to three-loop order, $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$, are calculated in the single-mass case. For the $_2F_1$-related master integrals of the problem, we use a semi-analytic method basedon series expansions and utilize the first-order differential equations for the master integrals whichdoes not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to $O(ep^5)$ in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable $x in ]0,infty[$ using highly precise series expansions to obtain the imaginary part of the physical amplitude for $x in ]0,1]$ at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-$x$ region. We also derive expansions in the region of small and large values of $x$. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.},
number = {24-02},
year = {2024},
month = {March},
note = {arXiv:2403.00513 [[hep-ph]},
keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, numerics},
length = {14},
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)}
}

2023

[Banerjee]

Invariants of the quartic binary form and proofs of Chen's conjectures on inequalities for the partition function and the Andrews' spt function

K. Banerjee

Submitted to the RISC Report Series. 2023. Licensed under CC BY 4.0 International. [pdf]
[bib]
@techreport{RISC6750,
author = {K. Banerjee},
title = {{Invariants of the quartic binary form and proofs of Chen's conjectures on inequalities for the partition function and the Andrews' spt function}},
language = {english},
abstract = {An extensive amount of study has been done on inequalities for the partition function, emerged primarily through works of Chen. In particular, the Tur'{a}n inequality and the higher order Tur'{a}n inequalities for $p(n)$ has been one of the most predominant theme. Among many others, one of the most notable one is Griffin, Ono, Rolen, and Zagier's result in which they proved that for every integer $d geq 1$, there exists an integer $N(d)$ such that the Jensen polynomial of degree $d$ and shift $n$ associated with the partition function, denoted by $J^{d,n}_p(x)$, has only distinct real roots for all $n geq N(d)$, earlier conjectured by Chen, Jia, and Wang and Ono independently. Later, Larson and Wagner have provided an estimate of upper bound for $N(d)$. This phenomena in turn implies that the discriminant of $J^{d,n}_p(x)$ is positive; i.e., $text{Disc}_{x}(J^{d,n}_p)>0$. For $d=2$, $text{Disc}_{x}(J^{2,n}_p)>0$ when $n geq N(2)=26$ is equivalent to the fact that $(p(n))_{n geq 26}$ is $log$-concave. In 2017, Chen undertook a comprehensive investigation on inequalities for $p(n)$ through the lens of invariant theory of binary forms of degree $n$. Positivity of the invariant of a quadratic binary form (resp. cubic binary form) associated with $p(n)$ reflects that the sequence $(p(n))_{n geq 26}$ satisfies the Tur'{a}n inequality (resp. $(p(n))_{n geq 95}$ satisfies the higher order Tur'{a}n inequality). Chen further studied on the two invariants for a quartic binary form where its coefficients are shifted values of integer partitions and conjectured four inequalities for $p(n)$. In this paper, we give explicit error bounds for the asymptotic expansion of the shifted partition function $p(n-ell)$ for any non-negative integer $ell$. As an application of these infinite family of inequalities, we confirm the conjectures of Chen. Moreover, three family of inequalities related to the partition function have been studied in this paper, namely, higher order Laguerre inequalities, higher order shifted differences, and higher order log-concavity. In context of higher order Laguerre inequalities for $p(n)$, we settle a conjecture of Wagner. For higher order shifted difference of $p(n)$, we extend a result of Gomez, Males, and Rolen. In context of higher order log-concavity for $p(n)$, we prove discuss on the asymptotic growth for the $r$-fold applications (with $rin {1,2,3}$) of the operator $mathcal{L}$ on $p(n)$ defined by $mathcal{L}(p(n))=p(n)^2-p(n-1)p(n+1)$ and propose a conjecture on infinite log-concavity in this regard. Furthermore, we will show how to construct a unified framework to prove partition function inequalities of the above types and discuss a few possible applications of such construction. Finally, we prove all the Chen's conjectures related to the inequalities for the Andrews' spt function, denoted by spt$(n)$, arising from invariants of quartic binary form using inequalities for the shifted partition function.},
year = {2023},
keywords = {the partition function, Andrews’ spt function, Hardy-Ramanujan-Rademacher for- mula, invariants of binary forms, combinatorial inequalities},
length = {55},
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)}
}
[Schneider]

Refined telescoping algorithms in $R\Pi\Sigma$-extensions to reduce the degrees of the denominators

C. Schneider

Technical report no. 23-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. arXiv:2302.03563 [cs.SC]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6682,
author = {C. Schneider},
title = {{Refined telescoping algorithms in $R\Pi\Sigma$-extensions to reduce the degrees of the denominators}},
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 $R\Pi\Sigma$-ring extensions that are built over general $\Pi\Sigma$-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.},
number = {23-01},
year = {2023},
month = {February},
note = {arXiv:2302.03563 [cs.SC]},
keywords = {telescoping, difference rings, reduced denominators, nested sums},
length = {18},
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]

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]
[bib]
@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)}
}
[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]
[bib]
@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)}
}
[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]
[bib]
@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)}
}
[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]
[bib]
@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)}
}
[STUDENT]

Formalisation of Relational Algebra and a SQL-like Language with the RISCAL Model Checker

Joachim Borya

Technical report no. 23-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2023. Bachelor thesis. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6706,
author = {Joachim Borya},
title = {{Formalisation of Relational Algebra and a SQL-like Language with the RISCAL Model Checker}},
language = {english},
abstract = {The relational database model is based on the mathematical concept of relational algebra.Query languages have been developed to make data available quickly without creatingdedicated access procedures that depend on the internal representation of the data. SQL(structured query language) can be seen as a quasi-standard for this. This thesis dealswith the formalization and verification of relational algebra and a small but elementarysubset of SQL with the help of the RISCAL model checker, a software tool for the formalspecification and verification of mathematical theories and algorithms.},
number = {23-06},
year = {2023},
month = {May},
keywords = {formal methods, program verification, model checking, automated theorem proving},
length = {77},
type = {Bachelor thesis},
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)}
}
[STUDENT]

Model Checking Concurrent Systems Under Fairness Constraints in RISCAL

Ágoston Sütő

Technical report no. 23-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2023. Master's thesis. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6708,
author = {Ágoston Sütő},
title = {{Model Checking Concurrent Systems Under Fairness Constraints in RISCAL}},
language = {english},
abstract = {Model checking is a method for verifying that a program satisfies certain desirable properties formalised using mathematical logic. It is a rigorous method, similar to theorem proving, but it is generally applied when theorem proving would be too difficult due to the complexity of the algorithm, such as in concurrent systems. Model checking is used in the software industry. RISCAL (RISC Algorithm Language) is a language and software system that can be used to describe algorithms over a finite domain, specify their behaviour and then validate the specification. While it mainly focuses on deterministic algorithms, it has limited support for non-deterministic systems as well.The thesis extends the support for non-deterministic systems in RISCAL by allowing the user to specify complex properties about their behaviour in the language of Linear Temporal Logic (LTL) and then to validate them. The core contribution is a model checker implemented in Java using the so-called automaton-based explicit state model checking approach. The software is capable of verifying certain properties that could not be handled by a well-known model checker used in the industry. While in most cases it has underperformed its competitors, our implementation is promising, especially when it comes to properties with certain side conditions, called fairness constraints. The majority of the thesis is be concerned with the theoretical aspects of the automaton-based model checking approach, which is followed by a description of the implementation and various benchmarks.},
number = {23-07},
year = {2023},
month = {May},
keywords = {formal methods, model checking, concurrent systems, nondeterminism, linear temporal logic},
sponsor = {Supported by Aktion Österreich–Slowakei project grant Nr. 2019-10-15-003 “Semantic Modeling of Component-Based Program Systems”},
length = {102},
type = {Master's thesis},
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)}
}
[de Freitas]

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

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

Technical report no. 23-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2023. arXiv:2307.02983 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6716,
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.},
number = {23-08},
year = {2023},
month = {July},
note = {arXiv:2307.02983 [hep-ph]},
keywords = {form factor, Feynman diagram, computer algebra, holonomic properties, difference equations, differential equations, symbolic summation, numerical matching, analytic continuation, guessing, PSLQ},
length = {92},
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)}
}
[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

Technical report no. 23-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2023. arXiv:2306.16550 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6714,
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}},
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.},
number = {23-09},
year = {2023},
month = {June},
note = {arXiv:2306.16550 [hep-ph]},
keywords = {deep-inelastic scattering, 3-loop Feynman diagrams, (inverse) Mellin transform, binomial sums},
length = {7},
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)}
}
[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]
[bib]
@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)}
}
[Schneider]

Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package

Johannes Bluemlein, Nikolai Fadeev, Carsten Schneider

Technical report no. 23-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2023. Appeared in ACM Communications in Computer Algebra, Vol. 57, No. 2, Issue 224, June 2023, arXiv:2308.06042 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6739,
author = {Johannes Bluemlein and Nikolai Fadeev and Carsten Schneider},
title = {{Computing Mellin representations and asymptotics of nested binomial sums in a symbolic way: the RICA package}},
language = {english},
abstract = {Nested binomial sums form a particular class of sums that arise in the context of particle physics computations at higher orders in perturbation theory within QCD and QED, but that are also mathematically relevant, e.g., in combinatorics. We present the package RICA (Rule Induced Convolutions for Asymptotics), which aims at calculating Mellin representations and asymptotic expansions at infinity of those objects. These representations are of particular interest to perform analytic continuations of such sums. },
number = {23-11},
year = {2023},
month = {August},
note = {Appeared in ACM Communications in Computer Algebra, Vol. 57, No. 2, Issue 224, June 2023, arXiv:2308.06042 [hep-ph]},
keywords = {Mellin transform, asymptotic expansions, nested sums, nested integrals, computer algebra},
length = {4},
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)}
}
[de Freitas]

The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$

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

Technical report no. 23-12 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). November 2023. arXiv:2311.00644 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6753,
author = {J. Ablinger and A. Behring and J. Bluemlein and A. De Freitas and A. von Manteuffel and C. Schneider and K. Schoenwald},
title = {{The first--order factorizable contributions to the three--loop massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$}},
language = {english},
abstract = {The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$ and $Delta A_{Qg}^{(3)}$contain first--order factorizable and non--first--order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first--order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color--$zeta$ factors for the cases in which also non--first--order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin $N$--space, and correspondingly, on Kummer--Poincar'e and square--root valued alphabets in Bjorken--$x$ space. We present a complete discussion of the possibilities of solving the present problem in $N$--space analytically and we also discuss the limitations in the present case to analytically continue the given $N$--space expressions to $N in mathbb{C}$ by strict methods. The representation through generating functions allows a well synchronized representation of the first--order factorizable results over a 17--letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in $x$--space, also containing up to weight {sf w = 5} special constants, which can be rationalized to Kummer--Poincar'e iterated integrals at special arguments. The analytic $x$--space representation requires separate analyses for the intervals $x in [0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large $x$ limits of the first--order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high--precision numerical solution of the problems.},
number = {23-12},
year = {2023},
month = {November},
note = {arXiv:2311.00644 [hep-ph]},
keywords = {Feynman diagram, massive operator matrix elements, computer algebra, differential equations, difference equations, coupled systems, nested integrals, nested sums},
length = {58},
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)}
}
[Schreiner]

The SLANG Semantics-Based Language Generator

Wolfgang Schreiner, William Steingartner

Technical report no. 23-13 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). September 2023. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6749,
author = {Wolfgang Schreiner and William Steingartner},
title = {{The SLANG Semantics-Based Language Generator}},
language = {english},
abstract = {This report documents the SLANG semantics-based language generator. SLANG is a software for generating rapid prototype implementations of programming languages from their formal specifications. Its input is a text file that describes the abstract syntax of a language and its concrete text representation; from this, a parser is generated (utilizing the ANTLR4 tool) that transforms the text representation of a program into its abstract syntax tree and a printer that generates from the abstract syntax tree its text representation. Furthermore, one can equip the language with a formal type system (by logical inference rules) from which a type checker is generated. Finally, one can give the language a formal semantics, in the denotational style (by function equations) and/or in the big-step operational style (by transition steps); from this, a language interpreter is generated. SLANG is implemented in Java and produces Java source code; it should be easy to extend the software also to other target languages.},
number = {23-13},
year = {2023},
month = {September},
keywords = {formal semantics of programming languages, denotational semantics, operational semantics, type systems, interpreters},
sponsor = {Supported by the Slovak Academic Information Agency SAIA project 2023-03-15-001 “Semantics-Based Rapid Prototyping of Domain-Specific Languages”},
length = {59},
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)}
}
[Paule]

MacMahon's Partition Analysis XV: Parity

G.E. Andrews and P. Paule

Technical report no. 23-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). December 2023. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6935,
author = {G.E. Andrews and P. Paule},
title = {{MacMahon's Partition Analysis XV: Parity}},
language = {english},
abstract = {We apply the methods of partition analysis to partitions in which the parity of parts plays a role. We begin with an in-depth treatment of the generating function for the partitions from the first G ̈ollnitz-Gordon identity. We then deduce a Schmidt-type theorem related to the false theta functions. We also consider: (1) position parity, (2) partitions with distinct even parts, (3) partitions with distinct odd parts. One of the corollaries of these last considerations is a new interpretation of Hei-Chi Chan’s cubic partitions. A second part of our article is devoted to the algorithmic derivation of identities and arithmetic congruences related to the generating functions considered in part one, including cubic partitions. To this end, Smoot’s implementation of Radu’s Ramanujan-Kolberg algorithm is used. Finally, we give a short description which explains how to use the Omega package to derive special instances of the results of part one.},
number = {23-14},
year = {2023},
month = {December},
keywords = {G ̈ollnitz-Gordon partitions, Schmidt-type identities, parity of parts, MacMahon’s partition analysis, q-series, partition congruences, Radu’s Ramanujan-Kolberg algorithm, the Omega package},
length = {31},
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)}
}

2022

[Schneider]

The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms

J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Technical report no. 22-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6487,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms}},
language = {english},
abstract = {We calculate the unpolarized and polarized two--loop massless off--shell operator matrix elements in QCD to $O(ep)$ in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations upto four--loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin--$N$ space or by harmonic polylogarithms in $z$--space. We also perform comparisons to the literature. },
number = {22-01},
year = {2022},
month = {February},
keywords = {QCD, Operator Matrix Element, 2-loop Feynman diagrams, computer algebra, large moment method},
length = {101},
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)}
}
[Nuspl]

Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences

P. Nuspl, V. Pillwein

Technical report no. 22-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6479,
author = {P. Nuspl and V. Pillwein},
title = {{Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences}},
language = {english},
abstract = {The class of $C^2$-finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients, i.e., coefficients satisfying a linear recurrence with constant coefficients themselves. Recently, we investigated computational properties of $C^2$-finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring.From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple $C^2$-finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function.},
number = {22-02},
year = {2022},
month = {February},
keywords = {difference equations, holonomic sequences, closure properties, generating functions, 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)}
}
[Schneider]

The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

J. Blümlein, C. Schneider

Technical report no. 22-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2022. arXiv:2203.13015 [hep-th]. Licensed under CC BY 4.0 International. [doi] [pdf]
[bib]
@techreport{RISC6495,
author = {J. Blümlein and C. Schneider},
title = {{The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals}},
language = {english},
abstract = {The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.},
number = {22-03},
year = {2022},
month = {March},
note = {arXiv:2203.13015 [hep-th]},
keywords = {Feynman integrals, computer algebra, special functions, linear differential equations, linear difference integrals},
length = {40},
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)}
}

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