On transcendental entire functions with infinitely many derivatives taking integer values at two points

Given a subset $S=\{s_0, s_1\}$ of the complex plane with two points and an infinite subset ${\mathscr S}$ of $S\times {\mathbb N}$, where ${\mathbb N}=\{0,1,2,\dots\}$ is the set of nonnegative integers, we ask for a lower bound for the order of growth of a transcendental entire function $f$ such that $f^{(n)}(s)\in{\mathbb Z}$ for all $(s,n)\in{\mathscr S}$. We first take ${\mathscr S}=\{s_0,s_1\}\times 2{\mathbb N}$, where $2{\mathbb N}=\{0,2,4,\dots\}$ is the set of nonnegative even integers. We prove that an entire function $f$ of sufficiently small exponential type such that $f^{(2n)}(s_0)\in{\mathbb Z}$ and $f^{(2n)}( s_1)\in{\mathbb Z}$ for all sufficiently large $n$ must be a polynomial. The estimate we reach is optimal, as we show by constructing a noncountable set of examples. The main tool, both for the proof of the estimate and for the construction of examples, is Lidstone polynomials. Our second example is $(\{s_0\}\times (2{\mathbb N}+1))\cup( \{ s_1\}\times 2{\mathbb N})$ (odd derivatives at $s_0$ and even derivatives at $s_1$). We use analogs of Lidstone polynomials which have been introduced by J.M.~Whittaker and studied by I.J.~Schoenberg. Finally, using results of W.~Gontcharoff, A. J.~Macintyre and J.M.~Whittaker, we prove lower bounds for the exponential type of a transcendental entire function $f$ such that, for each sufficiently large $n$, one at least of the two numbers $f^{(n)}(s_0)$, $f^{(n)}(s_1)$ is in ${\mathbb Z}$.