Related papers: Another Approach on Power Sums

We improve a result of Bennett concerning certain sequences involving sums of powers of positive integers.

A novel method of summation for power series is developed. The method is based on the self-similar approximation theory. The trick employed is in transforming, first, a series expansion into a product expansion and in applying the…

We obtain new bounds on complete rational exponential sums with sparse polynomials modulo a prime, under some mild conditions on the degrees of the monomials of such polynomials. These bounds, when they apply, give explicit versions of a…

This paper presents expressions for sums of powers of sine and cosine in terms of the basis for the field extension obtained by adjoining the sine or cosine to the field of rational numbers.

Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.

In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.

We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…

For any two arithmetic functions $f,g$ let $\bullet$ be the commutative and associative arithmetic convolution $(f\bullet g)(k):=\sum_{m=0}^k \left( \begin{array}{c} k m \end{array} \right)f(m)g(k-m)$ and for any $n\in\mathbb{N},$…

We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally…

In this paper we introduce a new approach to the concept of multipolynomials and generalize several results of the homogeneous polynomials and symmetric multilinear applications. We also present an abstract approach to the concept of…

Research on power values of power sums has gained much attention of late, partially due to the explosion of refinements in multiple advanced tools in (computational) Number Theory in recent years. In this survey, we present the key tools…

In the present paper a new mean value theorem for polynomials of special form is obtained. The case of sums on vertices of a regular polygon is studied. A criterion for a certain equation to be satisfied is obtained.

Sums of the form $\sum_{q \leq N_1 < \cdots < N_m \leq n}{a_{(m);N_m}\cdots a_{(2);N_2}a_{(1);N_1}}$ date back to the sixteen century when Vi\`ete illustrated that the relation linking the roots and coefficients of a polynomial had this…

We define an enumerative function F(n,k,P,m) which is a generalization of binomial coefficients. Special cases of this function are also power function, factorials, rising factorials and falling factorials. The first section of the paper is…

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$…

We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…

Let n,k be the positive integers, and let S_{k}(n) be the sums of the k-th power of positive integers up to n. By means of that we consider the evaluation of the sum of more general series by Bernstein polynomials. Additionally we show the…