Polynomial values of sums of hyperbolic binomial coefficients
type: Item
instance of: scholarly article
title: Polynomial values of sums of hyperbolic binomial coefficients
zbMATH Open document ID: 1480.11036
author: András Bazsó
author: Lajos Hajdu
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2021-05-05T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1572055510
review text: 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.
reviewed by: Maciej Ulas
Mathematics Subject Classification ID: 11D41
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11B37
zbMATH DE Number: 7344688
zbMATH Keywords: Pascal triangle
zbMATH Keywords: hyperbolic Pascal triangle
zbMATH Keywords: Chebyshev polynomials
zbMATH Keywords: recurrence sequences
zbMATH Keywords: polynomial values
zbMATH Keywords: power values
MaRDI profile type: Publication
OpenAlex ID: W2982269207
Wikidata QID: Q115478552
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DOI: 10.7169/FACM/1831
description: scientific article; zbMATH DE number 7344688