On the derivatives of the integer-valued polynomials (Q2285759)

Let \(n\) be a natural number and Int\(_n(\mathbb{Z})\) be the set of integer-valued polynomials of degree at most \(n.\) If we denote by \(c_{n,k}\) the smallest positive integer such that for any \(P\in \mathrm{Int}_n(\mathbb{Z}),\ c_{n,k}P^{(k)}\in \mathrm{Int}_n(\mathbb{Z}),\) Theorem 3.1 of the paper under review gives the form of \(c_{n,k}.\) In particular it is shown that \(c_n:=c_{n,1}=\mathrm{lcm}(1,2,\ldots,n).\) Furthermore, the smallest positive integer \(\lambda_n\) satisfying the property that for any \(P\in \mathrm{Int}_n(\mathbb{Z})\) and any \(k\in\mathbb{N}, \lambda_nP^{(k)}\in \mathrm{Int}_n(\mathbb{Z})\) is computed. Several applications are given.