Related papers: Sums of Powers and Majorization

We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of…

In which a review of the concept of countability is done in mathematics, subjecting review some of the theorems so far accepted, showing their inconsistency and also taking concrete elements on the countability of all the powers of the set…

Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…

We give necessary and sufficient conditions for majorization of realrooted polynomials sharing a common interlacer by means of residues coming from fraction decomposition. We also introduce a motivated notion called strong majorization, and…

We describe the limit zero distributions of sequences of polynomials with positive coefficients.

It is shown that if two hyperbolic polynomials have a particular factorization into quadratics, then their roots satisfy a power majorization relation whenever key coefficients in their factorizations satisfy a corresponding majorization…

In this article a new method of generating sums of like powers is presented.

We investigate the quantitative relationship between nonnegative polynomials and sums of squares of polynomials. We show that if the degree is fixed and the number of variables grows then there are significantly more nonnegative polynomials…

In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of…

We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.

We present new counterexamples, which provide stronger limitations to sums-differences statements than were previously known. The main idea is to consider non-uniform probability measures.

Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice…

Automatic sequences have many properties that other sequences (in particular, non-uniformly morphic sequences) do not necessarily share. In this paper we survey a number of different methods that can be used to prove that a given sequence…

We consider a generalisation of the classical Lehmer problem about the parity distribution of an integer and its modular inverse. We use some known estimates of exponential sums to study a more general question of simultaneous distribution…

We use an elementary argument to prove some finite sums involving expressions of the forms $(q)_n$ and $(a;q)_n$ along with inductive formulas for some sequences.

Binomial coefficients have been used for centuries in a variety of fields and have accumulated numerous definitions. In this paper, we introduce a new way of defining binomial coefficients as repeated sums of ones. A multitude of binomial…

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…

In this paper, we investigate two methods to express the natural powers of $2$ as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients…

Measures of voting power have been the subject of extensive research since the mid 1940s. More recently, similar measures of relative importance have been studied in other domains that include inconsistent knowledge bases, intensity of…