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

Abstract For each integer $$n\ge 1$$ n ≥ 1 we consider the unique polynomials $$P, Q\in {\mathbb {Q}}[x]$$ P , Q ∈ [ x ] of smallest degree n that are solutions equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$ ( ) + = . We derive numerous properties these and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, irreducibility results. also some related properties.