Polynomial values of (alternating) power sums
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.
scientific article; zbMATH DE number 6699132