Arithmetic properties of polynomial solutions of the Diophantine equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$

Bibliographic description

punktacja MEiN [2022]: 70

title:	Arithmetic properties of polynomial solutions of the Diophantine equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$
author:	Dilcher Karl, Ulas Maciej
journal title:	Periodica Mathematica Hungarica
volume:	84
date of publication :	2022
pages:	89-118
ISSN:	0031-5303
eISSN:	1588-2829
DOI:	10.1007/s10998-020-00376-5
language:	English
journal language:	English
abstract in English:	For each integer n≥1 we consider the unique polynomials P,Q∈Q[x] of smallest degree n that are solutions of the equation $P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties.
keywords in English:	recurrence sequence, polynomial Diophantine equation, discriminant, resultant, generating function
affiliation:	Wydział Matematyki i Informatyki : Instytut Matematyki
type:	journal article
subtype:	academic paper