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

TL;DR

This paper investigates the properties of specific polynomial solutions to a Diophantine equation, deriving explicit formulas, differential equations, and algebraic characteristics, revealing new insights into their structure and irreducibility.

Contribution

It introduces a detailed analysis of polynomial solutions to a particular Diophantine equation, including explicit formulas, differential equations, and algebraic properties, which were not previously known.

Findings

Explicit expansions and formulas for the polynomials.

Differential equations and recurrence relations derived.

Results on irreducibility and algebraic properties.

Abstract

For each integer $n\geq 1$ we consider the unique polynomials $P, Q\in\mathbb{Q}[x]$ 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.