Polynomial values of sums of hyperbolic binomial coefficients
Let \(\hat{s}_{n}(q)\) denote the sum of all elements in the \(n\)-th row of the hyperbolic Pascal triangle linked to \(\{4, q\}\) for \(q>5\). In the paper under review the authors investigate the Diophantine equation \(\hat{s}_{n}(q)=w^{l}\). The main result in the paper states that any \(q>5\) and \(w\in\mathbb{Z}\), the equation \(\hat{s}_{n}(q)=w^{l}\) has only finitely many solutions in positive integers \(n, l\). Further, we have \(\max\{n,l\}<C_{1}(q, w)\), where \(C_{1}(q, w)\) is an effectively computable constant depending only on \(q\) and \(w\). As an immediate consequence of the result the authors proved that for any \(q>5\), there are only finitely many rows of the hyperbolic Pascal triangle corresponding to \(\{4, q\}\) for which the sum of all elements is a power of any fixed integer \(w\). The authors obtain similar results in the case of the Diophantine equation \(\hat{s}_{n}(q)=ax^{l}+b\), where \(a, b\) are fixed rationals. There are many tools used in the proofs: from effective results of Shorey and Stewart concerning (almost) powers in linear recurrence sequences, through the tight connections of \(\hat{s}_{n}(q)\) with Chebyshev polynomials, to ineffective result of Bilu and Tichy.
scientific article; zbMATH DE number 7344688
scientific article