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Jakob Ablinger

Publications and source records attributed to Jakob Ablinger.

At least 19 recordsLinked to original sources
arXiv2021

Solving linear difference equations with coefficients in rings with idempotent representations

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 $RΠΣ$-extensions over $ΠΣ$-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.

cs.SC↗
arXiv2021

Extensions of the AZ-algorithm and the Package MultiIntegrate

We extend the (continuous) multivariate Almkvist-Zeilberger algorithm in order to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals over hyperexponential integrands and try to find closed form representations in terms of nested sums and products or iterated integrals. In addition, if we fail to compute a closed form solution in full generality, we may succeed in computing the first coefficients of the Laurent series expansions of such integrals in terms of indefinite nested 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) multivariate Almkvist Zeilberger algorithm to compute recurrences or differential equations for hyperexponential integrands and integrals. Together with the summation package Sigma and the package HarmonicSums our package provides methods to compute closed form representations (or coefficients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums or iterated integrals.

cs.SC↗
arXiv2019

qFunctions -- A Mathematica package for $q$-series and partition theory applications

We describe the qFunctions Mathematica package for $q$-series and partition theory applications. This package includes both experimental and symbolic tools. The experimental set of elements includes guessers for $q$-shift equations and recurrences for given $q$-series and fitting/finding explicit expressions for sequences of polynomials. This package can symbolically handle formal manipulations on $q$-differential, $q$-shift equations and recurrences, such as switching between these forms, finding the greatest common divisor of recurrences, and formal substitutions. Here, we also extend the classical method of the weighted words approach. Moreover, qFunctions has implementations that automate the recurrence system creation of the weighted words approach as well as a scheme on cylindric partitions.

cs.SC↗
arXiv2019

Proving two conjectural series for $ζ(7)$ and discovering more series for $ζ(7)$

We give a proof of two identities involving binomial sums at infinity conjectured by Z-W Sun. In order to prove these identities, we use a recently presented method i.e. we view the series as specializations of generating series and derive integral representations. Using substitutions, we express these integral representations in terms of cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive the results. These methods are implemented in the computer algebra package HarmonicSums.

math.CO↗
arXiv2019

Discovering and Proving Infinite Pochhammer Sum Identities

We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $π,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals or directly in terms of cyclotomic harmonic polylogarithms. Using substitutions, we express the root-valued iterated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants. The methods are implemented in the computer algebra package HarmonicSums.

math.CO↗
arXiv2018

Massive three loop form factors in the planar limit

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.

hep-ph↗
arXiv2017

Algebraic independence of sequences generated by (cyclotomic) harmonic sums

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.

cs.SC↗
arXiv2016

Heavy flavour corrections to polarised and unpolarised deep-inelastic scattering at 3-loop order

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.

hep-ph↗
arXiv2016

Algorithms to solve coupled systems of differential equations in terms of power series

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.

cs.SC↗
arXiv2016

Inverse Mellin Transform of Holonomic Sequences

We describe a method to compute the inverse Mellin transform of holonomic sequences, that is based on a method to compute the Mellin transform of holonomic functions. Both methods are implemented in the computer algebra package HarmonicSums.

cs.SC↗
arXiv2016

A toolbox to solve coupled systems of differential and difference equations

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 Poincaré iterated integrals.

cs.SC↗
arXiv2015

Discovering and Proving Infinite Binomial Sums Identities

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $π$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the interated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.

math.NT↗
arXiv2014

The package HarmonicSums: Computer Algebra and Analytic aspects of Nested Sums

This paper summarizes the essential functionality of the computer algebra package HarmonicSums. On the one hand HarmonicSums can work with nested sums such as harmonic sums and their generalizations and on the other hand it can treat iterated integrals of the Poincare and Chen-type, such as harmonic polylogarithms and their generalizations. The interplay of these representations and the analytic aspects are illustrated by concrete examples.

cs.SC↗
arXiv2014

Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams

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.

hep-th↗
arXiv2014

Calculating Massive 3-loop Graphs for Operator Matrix Elements by the Method of Hyperlogarithms

We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist $τ=2$ local operator insertions corresponding to spin $N$. They contribute to the massive operator matrix elements in QCD describing the massive Wilson coefficients for deep-inelastic scattering at large virtualities. Diagrams of this kind can be computed using an extended version to the method of hyperlogarithms, originally being designed for massless Feynman diagrams without operators. The method is applied to Benz- and $V$-type graphs, belonging to the genuine 3-loop topologies. In case of the $V$-type graphs with five massive propagators new types of nested sums and iterated integrals emerge. The sums are given in terms of finite binomially and inverse binomially weighted generalized cyclotomic sums, while the 1-dimensionally iterated integrals are based on a set of $\sim 30$ square-root valued letters. We also derive the asymptotic representations of the nested sums and present the solution for $N \in \mathbb{C}$. Integrals with a power-like divergence in $N$--space $\propto a^N, a \in \mathbb{R}, a > 1,$ for large values of $N$ emerge. They still possess a representation in $x$--space, which is given in terms of root-valued iterated integrals in the present case. The method of hyperlogarithms is also used to calculate higher moments for crossed box graphs with different operator insertions.

hep-ph↗
arXiv2013

Massive 3-loop Ladder Diagrams for Quarkonic Local Operator Matrix Elements

3-loop diagrams of the ladder-type, which emerge for local quarkonic twist-2 operator matrix elements, are computed directly for general values of the Mellin variable $N$ using Appell-function representations and applying modern summation technologies provided by the package {\sf Sigma} and the method of hyperlogarithms. In some of the diagrams generalized harmonic sums with $ξ\in \{1,1/2,2\}$ emerge beyond the usual nested harmonic sums. As the asymptotic representation of the corresponding integrals shows, the generalized sums conspire giving well behaved expressions for large values of $N$. These diagrams contribute to the 3-loop heavy flavor Wilson coefficients of the structure functions in deep-inelastic scattering in the region $Q^2 \gg m^2$.

hep-ph↗