Diophantine equations with truncated binomial polynomials

For positive integers k≤nk≤n let Pn,k(x):=∑j=0knjxj be the binomial expansion of (1+x)n(1+x)n truncated at the kkth stage. In this paper we show the finiteness of solutions of Diophantine equations of type Pn,k(x)=Pm,l(y)Pn,k(x)=Pm,l(y) in x,y∈Zx,y∈Z under assumption of irreducibility of truncated binomial polynomials Pn−1,k−1(x)Pn−1,k−1(x) and Pm−1,l−1(x)Pm−1,l−1(x). Although the irreducibility of Pn,k(x)Pn,k(x) has been studied by several authors, in general, this problem is still open. In addition, we give some results about the possible ways to write Pn,k(x)Pn,k(x) as a functional composition of two lower degree polynomials.