Related papers: Reduction of Multiple Harmonic Sums and Harmonic P…
Some years ago, the harmonic polynomial was introduced in order to understand better the harmonic topological index; for instance, it allows to obtain bounds of the harmonic index of the main products of graphs. Here, we obtain several…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
In this paper we investigate congruence relationships of particular finite generalized harmonic numbers sums. We suggest more transparent and simpler method to analyse these sums and present several additional results for certain special…
For each positive integer $m$, the $m$th order harmonic numbers are given by $$H_n^{(m)}=\sum_{0<k\le n}\frac1{k^m}\ \ (n=0,1,2,\ldots).$$ We discover exact values of some series involving harmonic numbers of order not exceeding four. For…
We study three classes of combinatorial sums involving central binomial coefficients and harmonic numbers, odd harmonic numbers, and even indexed harmonic numbers, respectively. In each case we use summation by parts to derive recursive…
The divergence of the harmonic series is proved by direct comparison with a series whose nth partial sum telescopes to the natural logarithm of n. The key idea is to apply the classical inequality x>=log(1+x) (valid for x>-1) with x=1/k and…
An alternative proof of the completeness of relational algebra with respect to allowed formulas of first-order logic is presented. The proof relies on the well-known embedding of relational algebra into cylindric algebra, which makes it…
A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated…
Given a connected non-negative unit form we construct an extended affine Lie algebra by giving a Chevalley basis for it. We also obtain this algebra as a quotient of an algebra defined by means of generalized Serre relations by M. Barot, D.…
We give two new constructions of the harmonic algebra of a lattice polytope $P$, a bigraded algebra whose character is the $q$-Ehrhart series of $P$ defined by Reiner and Rhoades. First, we show that the harmonic algebra is the associated…
We use elementary methods to establish three key recurrence relations: one for derangement numbers, a second for harmonic numbers, and a third for degenerate harmonic numbers. Our results not only contribute to the understanding of the…
We introduce an algebra model to study higher order sum rules for orthogonal polynomials on the unit circle. We build the relation between the algebra model and sum rules, and prove an equivalent expression on the algebra side for the sum…
The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc closure(D) generated by z and h --- where h is a…
The relationship between an algebra and its associated monomial algebra is investigated when at least one of the algebras is $d$-Koszul. It is shown that an algebra which has a reduced \grb basis that is composed of homogeneous elements of…
We present a common ground for infinite sums, unordered sums, Riemann/Lebesgue integrals, arc length and some generalized means. It is based on extending functions on finite sets using Hausdorff metric in a natural way.
A classification is given for (regular) positions of direct sums of two matroid algebras (unital algebraic limits of matrix algebras) in a matroid superalgebra, where the individual summands have index 2 in their associated corner algebra.…
Given $d \in \mathbb{N}$, we establish sum-product estimates for finite, non-empty subsets of $\mathbb{R}^d$. This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, non-empty set of $d…
The index of a Lie algebra is an important algebraic invariant, but it is notoriously difficult to compute. However, for the suggestively-named seaweed algebras, the computation of the index can be reduced to a combinatorial formula based…
There are many results for explicit expressions about $q$-multiple zeta values or $q$-harmonic sums on $A-\cdots-A$ indices, that is, the indices are the same. Though the way to treat $q$-multiple zeta values unless the indices are the…
The evaluation of the interaction between objects arranged on a lattice requires the computation of lattice sums. A scenario frequently encountered are systems governed by the Helmholtz equation in the context of electromagnetic scattering…