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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.

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.

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 $q^2 /m^2 \to \infty$ 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., Comment: 92 pages, 14 figures

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.

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.

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.

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., Comment: 42 pages, see also the overview article arXiv:2203.13011.v3: journal version

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments., Comment: 15 pages, overview article. v3: journal version

We introduce a general reduction strategy that enables one to search for solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposed by a direct sum of integral domains (using idempotent elements) that enjoys certain technical features and that the coefficients of the difference equation are not degenerated. Using this mechanism we can reduce the problem to find 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 obtain a general solver where the components of the linear difference equations and the solutions can be taken from difference rings that are 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 over roots of unity and nested sums defined over such objects.

A general overview of the existing difference ring theory for symbolic summation is given. Special emphasis is put on the user interface: the translation and back translation of the corresponding representations within the term algebra and the formal difference ring setting. In particular, canonical (unique) representations and their refinements in the introduced term algebra are explored by utilizing the available difference ring theory. Based on that, precise input-output specifications of the available tools of the summation package Sigma are provided., Comment: Various typos removed; improved presentation II

A non-trivial symbolic machinery is presented that can rephrase algorithmically a finite set of nested hypergeometric products in appropriately designed difference rings. As a consequence, one obtains an alternative representation in terms of one single product defined over a root of unity and nested hypergeometric products which are algebraically independent among each other. In particular, one can solve the zero-recognition problem: the input expression of nested hypergeometric products evaluates to zero if and only if the output expression is the zero expression. Combined with available symbolic summation algorithms in the setting of difference rings, one obtains a general machinery that can represent (and simplify) nested sums defined over nested products.

We present a complete algorithm that computes all hypergeometric solutions of homogeneous linear difference equations and rational solutions of parameterized linear difference equations in the setting of $\Pi\Sigma^*$-fields. More generally, we provide a flexible framework for a big class of difference fields that is built by a tower of $\Pi\Sigma^*$-field extensions over a difference field that satisfies certain algorithmic properties. As a consequence one can compute all solutions in terms of indefinite nested sums and products that arise within the components of a parameterized linear difference equation, and one can find all hypergeometric solutions that are defined over the arising sums and products of a homogeneous linear difference equation., Comment: Various typos have been removed and the presentation has been improved

Using symbolic summation tools in the setting of difference rings, we prove a two-parametric identity that relates rational approximations to $\zeta(4)$., Comment: 13 pages

In (S.B. Ekhad and D. Zeilberger, 2020) an exciting case study has been initiated in which experimental mathematics and symbolic computation are utilized to discover new properties concerning the so-called Absent-Minded Passengers Problem. Based on these results, Doron Zeilberger raised some challenging tasks to gain further probabilistic insight. In this note we report on this enterprise. In particular, we demonstrate how the computer algebra packages of RISC can be used to carry out the underlying heavy calculations., Comment: Removed various typos and inserted an extra link for a Mathematica notebook to repeat the (not-so-costly) calculations

The large moment method can be used to compute a large number of moments of physical quantities that are described by coupled systems of linear differential equations. Besides these systems the algorithm requires a certain number of initial values as input, that are often hard to derive in a preprocessing step.Thus a major challenge is to keep the number of initial values as small as possible. We present the basic ideas of the underlying large moment method and present refined versions that reduce significantly the number of required initial values.

We present results on the calculation of the polarized 2- and 3-loop anomalous dimensions in a massive computation of the associated operator matrix element. We also discuss the treatment of $\gamma_5$ and derive results in the M-scheme.10, Comment: 10 pages LATEX. arXiv admin note: substantial text overlap with arXiv:1908.03779

Recently, it has been shown constructively how a finite set of hypergeometric products, multibasic hypergeometric products or their mixed versions can be modeled properly in the setting of formal difference rings. Here special emphasis is put on robust constructions: whenever further products have to be considered, one can reuse --up to some mild modifications-- the already existing difference ring. In this article we relax this robustness criteria and seek for another form of optimality. We will elaborate a general framework to represent a finite set of products in a formal difference ring where the number of transcendental product generators is minimal. As a bonus we are able to describe explicitly all relations among the given input products.

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.

In perturbative calculations, e.g., in the setting of Quantum Chromodynamics (QCD) one aims at the evaluation of Feynman integrals. Here one is often faced with the problem to simplify multiple nested integrals or sums to expressions in terms of indefinite nested integrals or sums. Furthermore, one seeks for solutions of coupled systems of linear differential equations, that can be represented in terms of indefinite nested sums (or integrals). In this article we elaborate the main tools and the corresponding packages, that we have developed and intensively used within the last 10 years in the course of our QCD-calculations.

An overview is presented on the current status of main mathematical computation methods for the multi-loop corrections to single scale observables in quantum field theory and the associated mathematical number and function spaces and algebras. At present massless single scale quantities can be calculated analytically in QCD to 4-loop order and single mass and double mass quantities to 3-loop order, while zero scale quantities have been calculated to 5-loop order. The precision requirements of the planned measurements, particularly at the FCC-ee, form important challenges to theory, and will need important extensions of the presently known methods., Comment: 29 pages LATEX

We present the color planar and complete light quark QCD contributions to the three loop heavy quark form factors in the case of vector, axial-vector, scalar and pseudo-scalar currents. We evaluate the master integrals applying a new method based on differential equations for general bases, which is applicable for any first order factorizing systems. The analytic results are expressed in terms of harmonic polylogarithms and real-valued cyclotomic harmonic polylogarithms., Comment: 10 pages; Proceedings of the Loops and Legs in Quantum Field Theory, 29th April 2018 - 4th May 2018, St. Goar, Germany; Report number modified

Cyclotomic polylogarithms are reviewed and new results concerning the special constants that occur are presented. This also allows some comments on previous literature results using PSLQ.

An improved multi-summation approach is introduced and discussed that enables one to simultaneously handle indefinite nested sums and products in the setting of difference rings and holonomic sequences. Relevant mathematics is reviewed and the underlying advanced difference ring machinery is elaborated upon. The flexibility of this new toolbox contributed substantially to evaluating complicated multi-sums coming from particle physics. Illustrative examples of the functionality of the new software package RhoSum are given., Comment: Modified Proposition 2.1 and Corollary 2.1

In recent years, Karr's difference field theory has been extended to the so-called $R\Pi\Sigma$-extensions in which one can represent not only indefinite nested sums and products that can be expressed by transcendental ring extensions, but one can also handle algebraic products of the form $\alpha^n$ where $\alpha$ is a root of unity. In this article we supplement this summation theory substantially by the following building block. We provide new algorithms that represent a finite number of hypergeometric or mixed $(q_1,...,q_e)$-multibasic hypergeometric products in such a difference ring. This new insight provides a complete summation machinery that enables one to formulate such products and indefinite nested sums defined over such products in $R\Pi\Sigma$-extensions fully automatically. As a side-product, one obtains compactified expressions where the products are algebraically independent among each other, and one can solve the zero-recognition problem for such products., Comment: Removed various typos

We consider linear systems of recurrence equations whose coefficients are given in terms of indefinite nested sums and products covering, e.g., the harmonic numbers, hypergeometric products, $q$-hypergeometric products or their mixed versions. These linear systems are formulated in the setting of $\Pi\Sigma$-extensions and our goal is to find a denominator bound (also known as universal denominator) for the solutions; i.e., a non-zero polynomial $d$ such that the denominator of every solution of the system divides $d$. This is the first step in computing all rational solutions of such a rather general recurrence system. Once the denominator bound is known, the problem of solving for rational solutions is reduced to the problem of solving for polynomial solutions.

We device a new method to calculate a large number of Mellin moments of single scale quantities using the systems of differential and/or difference equations obtained by integration-by-parts identities between the corresponding Feynman integrals of loop corrections to physical quantities. These scalar quantities have a much simpler mathematical structure than the complete quantity. A sufficiently large set of moments may even allow the analytic reconstruction of the whole quantity considered, holding in case of first order factorizing systems. In any case, one may derive highly precise numerical representations in general using this method, which is otherwise completely analytic., Comment: 4 pages LATEX

We report on progress in the calculation of 3-loop corrections to the deep-inelastic structure functions from massive quarks in the asymptotic region of large momentum transfer $Q^2$. Recently completed results allow us to obtain the $O(a_s^3)$ contributions to several heavy flavour Wilson coefficients which enter both polarised and unpolarised structure functions for lepton-nucleon scattering. In particular, we obtain the non-singlet contributions to the unpolarised structure functions $F_2(x,Q^2)$ and $x F_3(x,Q^2)$ and the polarised structure function $g_1(x,Q^2)$. From these results we also obtain the heavy flavour contributions to the Gross-Llewellyn-Smith and the Bjorken sum rules., Comment: 16 pages, 5 figures; contribution to the proceedings of the QCD Evolution 2016 workshop, Amsterdam

Using integration by parts relations, Feynman integrals can be represented in terms of coupled systems of differential equations. In the following we suppose that the unknown Feynman integrals can be given in power series representations, and that sufficiently many initial values of the integrals are given. Then there exist algorithms that decide constructively if the coefficients of their power series representations can be given within the class of nested sums over hypergeometric products. In this article we will work out the calculation steps that solve this problem. First, we will present a successful tactic that has been applied recently to challenging problems coming from massive 3-loop Feynman integrals. Here our main tool is to solve scalar linear recurrences within the class of nested sums over hypergeometric products. Second, we will present a new variation of this tactic which relies on more involved summation technologies but succeeds in reducing the problem to solve scalar recurrences with lower recurrence orders. The article will work out the different challenges of this new tactic and demonstrates how they can be treated efficiently with our existing summation technologies.

We show that double sums of the form $$ \sum_{i,j=-n} ^{n} |i^sj^t(i^k-j^k)^\beta| \binom {2n} {n+i} \binom {2n} {n+j} $$ can always be expressed in terms of a linear combination of just four functions, namely $\binom {4n}{2n}$, ${\binom {2n}n}^2$, $4^n\binom {2n}n$, and $16^n$, with coefficients that are rational in $n$. We provide two different proofs: one is algorithmic and uses the second author's computer algebra package Sigma; the second is based on complex contour integrals. In many instances, these results are extended to double sums of the above form where $\binom {2n}{n+j}$ is replaced by $\binom {2m}{m+j}$ with independent parameter $m$., Comment: AmS-LaTeX, 42 pages; final version

The Novelli-Pak-Stoyanovskii algorithm is a sorting algorithm for Young tableaux of a fixed shape that was originally devised to give a bijective proof of the hook-length formula. We obtain new asymptotic results on the average case and worst case complexity of this algorithm as the underlying shape tends to a fixed limit curve. Furthermore, using the summation package Sigma we prove an exact formula for the average case complexity when the underlying shape consists of only two rows. We thereby answer questions posed by Krattenthaler and M\"uller.

Recently, $R\Pi\Sigma^*$-extensions have been introduced which extend Karr's $\Pi\Sigma^*$-fields substantially: one can represent expressions not only in terms of transcendental sums and products, but one can work also with products over primitive roots of unity. Since one can solve the parameterized telescoping problem in such rings, covering as special cases the summation paradigms of telescoping and creative telescoping, one obtains a rather flexible toolbox for symbolic summation. This article is the continuation of this work. Inspired by Singer's Galois theory of difference equations we will work out several alternative characterizations of $R\Pi\Sigma^*$-extensions: adjoining naively sums and products leads to an $R\Pi\Sigma^*$-extension iff the obtained difference ring is simple iff the ring can be embedded into the ring of sequences iff the ring can be given by the interlacing of $\Pi\Sigma^*$-extensions. From the viewpoint of applications this leads to a fully automatic machinery to represent indefinite nested sums and products in such $R\Pi\Sigma^*$-rings. In addition, we work out how the parameterized telescoping paradigm can be used to prove algebraic independence of indefinite nested sums. Furthermore, one obtains an alternative reduction tactic to solve the parameterized telescoping problem in basic $R\Pi\Sigma^*$-extensions exploiting the interlacing property., Comment: Numerous small corrections; a corrected proof of Lemma 2.22; extra requirements of part (1) of Theorem 6.7 have been inserted

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter $\ep$ (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t.\ $\ep$ and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package \texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma}, \texttt{HarmonicSums} and \texttt{OreSys}. In all applications the representation in $x$-space is obtained as an iterated integral representation over general alphabets, generalizing Poincar\'{e} iterated integrals.

An expression in terms of (cyclotomic) harmonic sums can be simplified by the quasi-shuffle algebra in terms of the so-called basis sums. By construction, these sums are algebraically independent within the quasi-shuffle algebra. In this article we show that the basis sums can be represented within a tower of difference ring extensions where the constants remain unchanged. This property enables one to embed this difference ring for the (cyclotomic) harmonic sums into the ring of sequences. This construction implies that the sequences produced by the basis sums are algebraically independent over the rational sequences adjoined with the alternating sequence., Comment: Updated the title in order to avoid confusion; inserted Lemma 5 to bring in more insight and supplemented Example 3 with further comments

We present an algebraic framework to represent indefinite nested sums over hypergeometric expressions in difference rings. In order to accomplish this task, parts of Karr's difference field theory have been extended to a ring theory in which also the alternating sign can be expressed. The underlying machinery relies on algorithms that compute all solutions of a given parameterized telescoping equation. As a consequence, we can solve the telescoping and creative telescoping problem in such difference rings., Comment: Some typos are removed

A summation framework is developed that enhances Karr's difference field approach. It covers not only indefinite nested sums and products in terms of transcendental extensions, but it can treat, e.g., nested products defined over roots of unity. The theory of the so-called $R\Pi\Sigma^*$-extensions is supplemented by algorithms that support the construction of such difference rings automatically and that assist in the task to tackle symbolic summation problems. Algorithms are presented that solve parameterized telescoping equations, and more generally parameterized first-order difference equations, in the given difference ring. As a consequence, one obtains algorithms for the summation paradigms of telescoping and Zeilberger's creative telescoping. With this difference ring theory one obtains a rigorous summation machinery that has been applied to numerous challenging problems coming, e.g., from combinatorics and particle physics., Comment: The environments are labelled differently, some parts are restructured, an index has been inserted, various typos are removed

The construction of Mellin-Barnes (MB) representations for non-planar Feynman diagrams and the summation of multiple series derived from general MB representations are discussed. A basic version of a new package AMBREv.3.0 is supplemented. The ultimate goal of this project is the automatic evaluation of MB representations for multiloop scalar and tensor Feynman integrals through infinite sums, preferably with analytic solutions. We shortly describe a strategy of further algebraic summation., Comment: Contribution to the proceedings of the Loops and Legs 2014 conference

Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss algorithms for converting between sum and integral representations, mainly relying on the Mellin transform. To aid the conversion we worked out dedicated rewrite rules, based on which also some general patterns emerging in the process can be obtained., Comment: 13 pages LATEX, one style file, Proceedings of Loops and Legs in Quantum Field Theory -- LL2014,27 April 2014 -- 02 May 2014 Weimar, Germany