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

For each integer $n\ge 1$ we consider the unique polynomials $P, Q\in$\mathbb{Q}$[x]$ofsmallestdegree$n$thataresolutionsoftheequation$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$. 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.