Related papers: Harmonic sums and polylogarithms at non-positive m…
The generalized Dehn-Sommerville relations determine the odd subalgebra of the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs that satisfy the generalized Dehn-Sommerville relations for the combinatorial Hopf…
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…
We present here a large collection of harmonic and quadratic harmonic sums, that can be useful in applied questions, e.g., probabilistic ones. We find closed-form formulae, that we were not able to locate in the literature.
Relations among integrals of logarithms, polylogarithms and Euler sums are presented. A unifying element being the introduction of Nielsen's generalized polylogarithms.
We study noncommutative versions of holomorphic and harmonic functions on the unit disk.
In this paper we derive two expressions for the Hurwitz zeta function involving the complete Bell polynomials in the restricted case where q is a positive integer greater than 1. The arguments of the complete Bell polynomials comprise the…
We show some of the mathematics that is being developed for the computation of deep inelastic structure functions to three loops. These include harmonic sums, harmonic polylogarithms and a class of difference equations that can be solved…
We observe that the classical Faulhaber's theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression $a+b, a+2b, ..., a+nb$ is a polynomial in $na+n(n+1)b/2$.…
We survey various results and conjectures concerning multiple polylogarithms and the multiple zeta function. Among the results, we announce our resolution of several conjectures on multiple zeta values. We also provide a new integral…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
Thom polynomial describes the cohomology class Poincar\'e dual to the locus of particular singularity of a generic holomorphic map. In this paper we derive a closed formula for the generating function of its coefficients. The method is…
In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear…
We introduce the concept of disjunctive sum of squares for certifying nonnegativity of polynomials. Unlike the popular sum of squares approach where nonnegativity is certified by a single algebraic identity, the disjunctive sum of squares…
We present the applications of methods from nonlinear local harmonic analysis for calculations in nonlinear collective dynamics described by different forms of Vlasov-Maxwell-Poisson equations. Our approach is based on methods provided the…
The paper explores the possibility for summing Fourier series nonlinearly via the Pythagorean harmonic mean. It reports on new results for this summability with introduction of new concepts like the smoothing operator and semi-harmonic…
We develop the theory of multiple polylogarithms from analytic, Hodge and motivic point of view. Define the category of mixed Tate motives over a ring of integers in a number field. Describe explicitly the multiple polylogarithm Hopf…
We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive values of the…
An identity involving symmetric sums of regularized multiple zeta-star values of harmonic type was proved by Hoffman. In this paper, we prove an identity of shuffle type. We use Bell polynomials appearing in the study of set partitions to…
To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra…
In this paper, we propose new algorithms for finding a common point of the solution set of a pseudomonotone equilibrium problem and the set of fixed points of a symmetric generalized hybrid mapping in a real Hilbert space. The convergence…