A Difference Ring Theory for Symbolic Summation

Abstract

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 $R\Pi {\Sigma }^{\ast }$-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.

The environments are labelled differently, some parts are restructured, an index has been inserted, various typos are removed

References in corpus (7)

• A Refined Difference Field Theory for Symbolic Summation
• The 3-Loop Non-Singlet Heavy Flavor Contributions and Anomalous Dimensions for the Structure Function $F_2(x,Q^2)$ and Transversity
• The Transition Matrix Element A_{gq}(N) of the Variable Flavor Number Scheme at O(α_s^3)
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• The $O({\alpha }_s^3{n}_f{T}_F^2{C}_{A,F})$} Contributions to the Gluonic Massive Operator Matrix Elements
• A Symbolic Summation Approach to Feynman Integral Calculus
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Cited by in corpus (8)

• The 3-Loop Pure Singlet Heavy Flavor Contributions to the Structure Function $F_2(x,Q^2)$ and the Anomalous Dimension
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• Representing ($q$--)hypergeometric products and mixed versions in difference rings
• A toolbox to solve coupled systems of differential and difference equations
• Higher Order Heavy Quark Corrections to Deep-Inelastic Scattering
• Denominator Bounds for Systems of Recurrence Equations using $\Pi \Sigma$-Extensions
• 3-Loop Corrections to the Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering
• 3-loop Massive $O(T_F^2)$ Contributions to the DIS Operator Matrix Element $A_{gg}$