## RISC Reports Series

[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]
@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]
@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]
@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)}
}
[Pau]

### A framework for approximate generalization in quantitative theories

#### Temur Kutsia, Cleo Pau

Technical report no. 22-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6505,
author = {Temur Kutsia and Cleo Pau},
title = {{A framework for approximate generalization in quantitative theories}},
language = {english},
abstract = {Anti-unification aims at computing generalizations for given terms, retaining their common structure and abstracting differences by variables. We study quantitative anti-unification where the notion of the common structure is relaxed into "proximal'' up to the given degree with respect to the given fuzzy proximity relation. Proximal symbols may have different names and arities. We develop a generic set of rules for computing minimal complete sets of approximate generalizations and study their properties. Depending on the characterizations of proximities between symbols and the desired forms of solutions, these rules give rise to different versions of concrete algorithms.},
number = {22-04},
year = {2022},
month = {May},
keywords = {Generalization, anti-unification, quantiative theories, fuzzy proximity relations},
length = {22},
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]

### A comparison of algorithms for proving positivity of linearly recurrent sequences

#### P. Nuspl, V. Pillwein

Technical report no. 22-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6514,
author = {P. Nuspl and V. Pillwein},
title = {{A comparison of algorithms for proving positivity of linearly recurrent sequences}},
language = {english},
abstract = {Deciding positivity for recursively defined sequences based on only the recursive description as input is usually a non-trivial task. Even in the case of $C$-finite sequences, i.e., sequences satisfying a linear recurrence with constant coefficients, this is only known to be decidable for orders up to five. In this paper, we discuss several methods for proving positivity of $C$-finite sequences and compare their effectiveness on input from the Online Encyclopedia of Integer Sequences (OEIS).},
number = {22-05},
year = {2022},
month = {May},
keywords = {Difference equations Inequalities Holonomic sequences},
length = {17},
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]

### $C$-finite and $C^2$-finite Sequences in SageMath

#### P. Nuspl

Technical report no. 22-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6516,
author = {P. Nuspl},
title = {{$C$-finite and $C^2$-finite Sequences in SageMath}},
language = {english},
abstract = {We present the SageMath package rec_sequences which provides methods to compute with sequences satisfying linear recurrences. The package can be used to show inequalities of $C$-finite sequences, i.e., sequences satisfying a linear recurrence relation with constant coefficients. Furthermore, it provides functionality to compute in the $C^2$-finite sequence ring, i.e., to compute closure properties of sequences satisfying a linear recurrence with $C$-finite coefficients.},
number = {22-06},
year = {2022},
month = {June},
keywords = {Difference equations, Closure properties, Inequalities},
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)}
}
[Schreiner]

### The RISCTP Theorem Proving Interface - Tutorial and Reference Manual (Version 1.0.*)

#### Wolfgang Schreiner

Technical report no. 22-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6517,
author = {Wolfgang Schreiner},
title = {{The RISCTP Theorem Proving Interface - Tutorial and Reference Manual (Version 1.0.*)}},
language = {english},
abstract = {This report documents the RISCTP theorem proving interface. RISCTP consists of alanguage for specifying proof problems and of an associated software for solving theseproblems. The RISCTP language is a typed variant of first-order logic whose level ofabstraction is between that of higher level formal specification languages (such as thelanguage of the RISCAL model checker) and lower level theorem proving languages (such asthe language SMT-LIB supported by various satisfiability modulo theories solvers such as Z3).Thus the RISCTP language can serve as an intermediate layer that simplifies the connectionof specification and verification systems to theorem provers; in fact, it was developed toequip the RISCAL model checker with theorem proving capabilities. The RISCTP softwareis implemented in Java with an API that enables the implementation of such connections;however, RISCTP also provides a text-based frontend that allows its use as a theorem proveron its own. RISCTP already implements a backend that translates a proving problem intoSMT-LIB and solves it by the "black box" application of Z3; in the future, RISCTP shall alsoprovide built-in proving capabilities with greater transparency.},
number = {22-07},
year = {2022},
month = {June},
keywords = {automated reasoning, theorem proving, model checking, first-order logic, RISCAL, SMT-LIB, Z3},
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)}
}
[Schneider]

### Computer Algebra and Hypergeometric Structures for Feynman Integrals

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

Technical report no. 22-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2022. arXiv:2207.08524 [math-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6524,
author = {J. Bluemlein and M. Saragnese and C. Schneider},
title = {{Computer Algebra and Hypergeometric Structures for Feynman Integrals}},
language = {english},
abstract = {We present recent computer algebra methods that support the calculations of (multivariate) series solutions for (certain coupled systems of partial) linear differential equations. The summand of the series solutions may be built by hypergeometric products and more generally by indefinite nested sums defined over such products. Special cases are hypergeometric structures such as Appell-functions or generalizations of them that arise frequently when dealing with parameter Feynman integrals.},
number = {22-08},
year = {2022},
month = {July},
note = { arXiv:2207.08524 [math-ph]},
keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},
length = {11},
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 3-loop anomalous dimensions from off-shell operator matrix elements

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

Technical report no. 22-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6525,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The 3-loop anomalous dimensions from off-shell operator matrix elements}},
language = {english},
abstract = {We report on the calculation of the three--loop polarized and unpolarized flavor non--singlet and the polarized singlet anomalous dimensions using massless off--shell operator matrix elements in a gauge--variant framework. We also reconsider the unpolarized two--loop singlet anomalous dimensions and correct errors in the foregoing literature.},
number = {22-09},
year = {2022},
month = {July},
keywords = {hypergeometric series, Appell-series, symbolic summation, coupled systems, partial linear difference equations},
length = {12},
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]

### Truncated Hermite polynomials

#### Diego Dominici and Francisco Marcell{\'a}n

Technical report no. 22-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6531,
author = {Diego Dominici and Francisco Marcell{\'a}n},
title = {{Truncated Hermite polynomials}},
language = {english},
abstract = {We define the family of truncated Hermite polynomials $P_{n}left(x;zright)$, orthogonal with respect to the linear functional[Lleft[ pright] = int_{-z}^{z} pleft( xright) e^{-x^{2}} ,dx. ]The connection of $P_{n}left( x;zright)$ with the Hermite and Rys polynomialsis stated. The semiclassical character of $P_{n}left( x;zright)$ aspolynomials of class $2$ is emphasized.As a consequence, several properties of $P_{n}left( x;zright)$ concerningthe coefficients $gamma_{n}left( zright)$ in the three-term recurrencerelation they satisfy as well as the moments and the Stieltjes function of $L$are given. Ladder operators associated with the linear functional $L$, aholonomic differential equation (in $x)$ for the polynomials $P_{n}left(x;zright)$, and a nonlinear ODE for the functions $gamma_{n}left(zright)$ are deduced. },
number = {22-10},
year = {2022},
month = {August},
keywords = {Orthogonal polynomials, Gaussian distribution},
length = {37},
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]

### Comparative asymptotics for discrete semiclassical orthogonal polynomials

#### Diego Dominici

Technical report no. 22-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6579,
author = {Diego Dominici},
title = {{Comparative asymptotics for discrete semiclassical orthogonal polynomials}},
language = {english},
abstract = {We study the ratio $frac{P_{n}left( x;zright) }{phi_{n}left( xright)}$ asymptotically as $nrightarrowinfty,$ where the polynomials $P_{n}left(x;zright)$ are orthogonal with respect to a discrete linear functional and$phi_{n}left( xright)$ denote the falling factorial polynomials.We give recurrences that allow the computation of high order asymptoticexpansions of $P_{n}left( x;zright)$ and give examples for most discretesemiclassical polynomials of class $sleq2.$We show several plots illustrating the accuracy of our results.},
number = {22-11},
year = {2022},
month = {August},
keywords = {Orthogonal polynomials, asymptotic analysis },
length = {53},
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 massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2, F_L, xF_3$ and $g_1$

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

Technical report no. 22-12 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2022. arXiv:2208.14325 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6583,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_2, F_L, xF_3$ and $g_1$}},
language = {english},
abstract = {We calculate the massless unpolarized Wilson coefficients for deeply inelastic scattering for thestructure functions $F_2(x,Q^2), F_L(x,Q^2), x F_3(x,Q^2)$ in the $overline{sf MS}$ scheme and the polarized Wilson coefficients of the structure function $g_1(x,Q^2)$ in the Larin scheme up to three--loop order in QCD in a fully automated way based on the method of arbitrary high Mellin moments. We workin the Larin scheme in the case of contributing axial--vector couplings or polarized nucleons. For the unpolarized structure functions we compare to results given in the literature. The polarized three--loop Wilson coefficients are calculated for the first time. As a by--product we also obtain the quarkonic three--loop anomalous dimensions from the $O(1/ep)$ terms of the unrenormalized forward Compton amplitude. Expansions for small and large values of the Bjorken variable $x$ are provided.},
number = {22-12},
year = {2022},
month = {August},
note = {arXiv:2208.14325 [hep-ph]},
keywords = {massless unpolarized Wilson coefficients, large moment method, linear difference equations, computer algebra,coupled systems of linear differential equations},
length = {160},
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]

### Error bounds for the asymptotic expansion of the partition function

#### Koustav Banerjee, Peter Paule, Cristian-Silviu Radu, Carsten Schneider

Technical report no. 22-13 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). September 2022. arXiv:2209.07887 [math.NT]. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6620,
author = {Koustav Banerjee and Peter Paule and Cristian-Silviu Radu and Carsten Schneider},
title = {{Error bounds for the asymptotic expansion of the partition function}},
language = {english},
abstract = {Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion for $p(n)$ with an explicit error bound is not known. Recently O'Sullivan studied the asymptotic expansion of $p^{k}(n)$-partitions into $k$th powers, initiated by Wright, and consequently obtained an asymptotic expansion for $p(n)$ along with a concise description of the coefficients involved in the expansion but without any estimation of the error term. Here we consider a detailed and comprehensive analysis on an estimation of the error term obtained by truncating the asymptotic expansion for $p(n)$ at any positive integer $n$. This gives rise to an infinite family of inequalities for $p(n)$ which finally answers to a question proposed by Chen. Our error term estimation predominantly relies on applications of algorithmic methods from symbolic summation. },
number = {22-13},
year = {2022},
month = {September},
note = {arXiv:2209.07887 [math.NT]},
keywords = {partition function, asymptotic expansion, error bounds, symbolic summation},
length = {43},
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 XIV: Partitions with n copies of n

#### G.E. Andrews, P. Paule

Technical report no. 22-14 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). October 2022. Research Article. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6626,
author = {G.E. Andrews and P. Paule},
title = {{MacMahon's Partition Analysis XIV: Partitions with n copies of n}},
language = {english},
abstract = {We apply the methods of partition analysis to partitions with~$n$ copies of~$n$. This allows us to obtain multivariable generatingfunctions related to classical Rogers-Ramanujan type identities. Also, partitionswith $n$ copies of $n$ are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partitioncongruences.},
number = {22-14},
year = {2022},
month = {October},
keywords = {partitions, overpartitions, partitions with $n$ copies of $n$, partition analysis, $q$-series, modular forms and partition congruences, Radu's Ramanujan-Kolberg algorithm},
length = {30},
type = {Research Article},
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 Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$

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

Technical report no. 22-15 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). November 2022. arXiv:2211.05462 [hep-ph]. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6629,
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 = {{The Unpolarized and Polarized Single-Mass Three-Loop Heavy Flavor Operator Matrix Elements $A_{gg, Q}$ and $Delta A_{gg, Q}$}},
language = {english},
abstract = {We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, $A_{gg,Q}(x,mu^2)$ and $Delta A_{gg,Q}(x,mu^2)$, at three--loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin $N$--space and iterated integrals over square root--valued alphabets in momentum fraction $x$--space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction $x$--space and derive numerical results. The present results complete the gluon transition matrix elements both of the single-- and double--mass variable flavor number scheme to three--loop order.},
number = {22-15},
year = {2022},
month = {November},
note = {arXiv:2211.05462 [hep-ph]},
keywords = {Feynman integrals, linear difference equations, linear differential equations, binomial sums, harmonic sums, iterative integrals, computer algebra},
length = {48},
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]

### Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials

#### Diego Dominici and Juan José Moreno Balcázar

Technical report no. 22-16 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). November 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6625,
author = {Diego Dominici and Juan José Moreno Balcázar},
title = {{Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials}},
language = {english},
abstract = {In this paper we tackle the asymptotic behaviour of a family of orthogonalpolynomials with respect to a nonstandard inner product involving the forwardoperator $Delta$. Concretely, we treat the generalized Charlier weights inthe framework of $Delta$--Sobolev orthogonality. We obtain an asymptoticexpansion for this orthogonal polynomials where the falling factorialpolynomials play an important role.},
number = {22-16},
year = {2022},
month = {November},
keywords = {Sobolev orthogonal polynomials, asymptotic analysis, discrete semiclassical orthogonal polynomials},
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)}
}

[Schneider]

### The Polarized Transition Matrix Element $A_{g, q}(N)$ of the Variable Flavor Number Scheme at $O(\alpha_s^3)$

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

Technical report no. 21-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2021. Available also under arXiv:2101.05733 [hep-ph]. [doi] [pdf]
@techreport{RISC6250,
author = {A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and K. Schönwald and and C. Schneider},
title = {{The Polarized Transition Matrix Element $A_{g,q}(N)$ of the Variable Flavor Number Scheme at $O(\alpha_s^3)$}},
language = {english},
abstract = {We calculate the polarized massive operator matrix element $A_{gq}^3(N)$ to 3-loop order inQuantum Chromodynamics analytically at general values of the Mellin variable $N$ both inthe single- and double-mass case in the Larin scheme. It is a transition function requiredin the variable flavor number scheme at $O(\alpha_s^3)$. We also present the results in momentumfraction space.},
number = {21-01},
year = {2021},
month = {January},
note = {Available also under arXiv:2101.05733 [hep-ph]},
keywords = {polarized massive operator matrix element, symbolic summation, harmonic sums, harmonic polylogarithm},
length = {25},
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)}
}
[Ablinger]

### Extensions of the AZ-algorithm and the Package MultiIntegrate

#### J. Ablinger

Technical report no. 21-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2021. [doi] [pdf]
@techreport{RISC6272,
author = {J. Ablinger},
title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},
language = {english},
abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate,can be considered as an enhanced implementation of the (continuous) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},
number = {21-02},
year = {2021},
month = {January},
keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},
length = {25},
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]

### Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

#### C. Schneider

Technical report no. 21-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2021. [doi] [pdf]
@techreport{RISC6280,
author = {C. Schneider},
title = {{Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation}},
language = {english},
abstract = {A general overview of the existing difference ring theory for symbolicsummation is given. Special emphasis is put on the user interface: the translationand back translation of the corresponding representations within the term algebra andthe formal difference ring setting. In particular, canonical (unique) representationsand their refinements in the introduced term algebra are explored by utilizing theavailable difference ring theory. Based on that, precise input-output specifications ofthe available tools of the summation package Sigma are provided.},
number = {21-03},
year = {2021},
month = {February},
keywords = {term algebra, canonical representations, difference rings, indefinite nested sums, symbolic summation},
length = {55},
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]

### Solving linear difference equations with coefficients in rings with idempotent representations

#### J. Ablinger, C. Schneider

Technical report no. 21-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2021. [doi] [pdf]
@techreport{RISC6284,
author = {J. Ablinger and C. Schneider},
title = {{Solving linear difference equations with coefficients in rings with idempotent representations}},
language = {english},
abstract = {We introduce a general reduction strategy that enables one to searchfor solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposedby a direct sum of integral domains (using idempotent elements)that enjoys certain technical features and that the coeicients ofthe difference equation are not degenerated. Using this mechanismwe can reduce the problem to ind solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains.Utilizing existing solvers in this integral domain setting, we obtaina general solver where the components of the linear differenceequations and the solutions can be taken from difference rings thatare built e.g., by $R\Pi\Sigma$-extensions over $\Pi\Sigma$-fields. This class of difference rings contains, e.g., nested sums and products, products overroots of unity and nested sums defined over such objects.},
number = {21-04},
year = {2021},
month = {February},
keywords = {linear difference equations, difference rings, idempotent elements},
length = {8},
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)}
}