Polynomial values of (alternating) power sums
type: Item
instance of: scholarly article
title: Polynomial values of (alternating) power sums
zbMATH Open document ID: 1374.11051
author: András Bazsó
published in: Acta Mathematica Hungarica
publication date: 2017-03-31T00:00:00Z
full work available at URL: PolValPowSums2015.pdf
review text: The author considers diophantine problems related to power sums and alternating power sums. A general power sum has the form \[ S_{a,b}^k(x)=b^k+(a+b)^k+\cdots+(a(x-1)+b)^k, \] while a general alternating power sum is definded as \[ T_{a,b}^k(x)=b^k-(a+b)^k+(2a+b)^k-\cdots+(-1)^{x-1}(a(x-1)+b)^k. \] \(S_{a,b}^k(x)\) and \(T_{a,b}^k(x)\) can be expressed by the Bernoulli and Euler polynomials, respectively: \[ S_{a,b}^k(x)=\frac{a^k}{k+1}\left(B_{k+1}\left(x+\frac{b}{a}\right)-B_{k+1}\left(\frac{b}{a}\right)\right), \] \[ T_{a,b}^k(x)=\frac{a^k}{2}\left(E_{k}\left(\frac{b}{a}\right)+(-1)^{x-1}E_{k}\left(x+\frac{b}{a}\right)\right). \] Note that the latter expression is not a polynomial in \(x\), but the two functions \[ T_{a,b}^{k\pm}(x)=\frac{a^k}{2}\left(E_{k}\left(\frac{b}{a}\right)\pm E_{k}\left(x+\frac{b}{a}\right)\right) \] are already polynomials. The diophantine problem studied in the paper is the following. Let us fix a rational polynomial \(g(y)\) of degree at least 3, and then we are looking for integers \(x\) and \(y\) such that \[ S_{a,b}^k(x)=g(y), \] and, similarly, \[ T_{a,b}^{k\pm}(x)=g(y). \] The author proves that, apart from some special cases, both of these equations have only finitely many solutions. The results are ineffective in the sense that specific bound on the solutions is not provided.
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 6699132
zbMATH Keywords: Diophantine equation
zbMATH Keywords: Bernoulli polynomial
zbMATH Keywords: Euler polynomial
reviewed by: István Mező
MaRDI profile type: Publication
OpenAlex ID: W2185613410
cites work: On alternating power sums of arithmetic progressions
cites work: On equal values of power sums of arithmetic progressions
cites work: A refinement of Faulhaber's theorem concerning sums of powers of natural numbers
cites work: A superelliptic equation involving alternating sums of powers
cites work: On the Diophantine equation $1^k+2^k+\dotsb+x^k=y^n$
cites work: Quadratic factors of f(x) -g(y)
cites work: The Diophantine equation f(x) = g(y)
cites work: Q4533267
cites work: On the Euler and Bernoulli polynomials.
cites work: On some generalizations of the diophantine equation $1^k + 2^k + ... + x^k = y^z$
cites work: Q3708856
cites work: Q4432060
cites work: On the equation $1^k + 2^k + ... + x^k = y^z$
cites work: Q4888639
cites work: A computational approach for solving $y^2=1^k+2^k+\dotsb+x^k$
cites work: Diophantine equations with Bernoulli polynomials
cites work: A class of Diophantine equations involving Bernoulli polynomials
cites work: Q5188092
cites work: Q3023603
cites work: On the simple zeros of shifted Euler polynomials
cites work: Diophantine equations with Euler polynomials
cites work: The equation \(1^p+2^p+3^p+\ldots+n^p=m^q\)
cites work: Q4488096
cites work: On the equation $f(1)1^k + f(2)2^k + ... + f(x)x^k + R(x)= by^z$
cites work: Q4301630
cites work: Q4884992
cites work: On the Diophantine equation \(1^k+2^k+\dots +x^k+R(x)=y^2\)
DOI: 10.1007/S10474-015-0493-X
description: scientific article; zbMATH DE number 6699132