Related papers: Generalized Harmonic, Cyclotomic, and Binomial Sum…
Numerical evaluations of Feynman integrals often proceed via a deformation of the integration contour into the complex plane. While valid contours are easy to construct, the numerical precision for a multi-loop integral can depend…
We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show…
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…
Explicit determinations of several classes of trigonometric sums are given. These sums can be viewed as analogues or generalizations of Gauss sums. In a previous paper, two of the present authors considered primarily sine sums associated…
We prove and generalize some recent conjectures of Z.-W. Sun on infinite series whose summands involve products of harmonic numbers and several binomial coefficients. We evaluate various classes of infinite sums in closed form by…
We describe an effective method for calculating certain infinite sums, generalizations of the classical Bernoulli polynomials. As shown by Edward Witten in his papers on two-dimensional gauge theories, the correlation functions of…
The $\varepsilon$-form of a system of differential equations for Feynman integrals has led to tremendeous progress in our abilities to compute Feynman integrals, as long as they fall into the class of multiple polylogarithms. It is…
Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated…
We introduce a generalization of the Stirling numbers via symmetric functions involving two weight functions. The resulting extension unifies previously known Stirling-type sequences with known symmetric function forms, as well as other…
In this paper, we mainly show that Euler sums of generalized hyperharmonic numbers can be expressed in terms of linear combinations of the classical Euler sums.
We introduce and study the generalized cyclotomic polynomials $\Phi_{A,S,n}(x)$ associated with a regular system $A$ of divisors and an arbitrary set $S$ of positive integers. We show that all of these polynomials have integer coefficients,…
Contents: Generalities, Chiral supermultiplets, Super Yang-Mills theory, Superspace Feynman graphs, Renormalization, Supercurrent, Finite theories.
In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic representations of Euler sums through values of polylogarithm function and Riemann zeta…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
We review recent progress that we have achieved in evaluating the class of fully massive vacuum integrals at five loops. After discussing topics that arise in classification, evaluation and algorithmic codification of this specific set of…
We introduce the multivariable connected sum which is a generalization of Seki-Yamamoto's connected sum and prove the fundamental identity for these sums by series manipulation. This identity yields explicit procedures for evaluating…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own…
Multi-loop Feynman integrals are key objects for the high-order correction computations in high energy phenomenology. These integrals with multiple scales, may have complicated symbol structures. We show that the dual conformal symmetry…
We discuss a progress in calculation of Feynman integrals which has been done with help of the differential equation method and demonstrate the results for a class of two-point two-loop diagrams.