  Let $s_0,s_1,\dots,s_{m-1}$ be complex numbers and $r_0,\dots,r_{m-1}$
rational integers in the range $0\le r_j\le m-1$. Our first goal is to prove
that if an entire function $f$ of sufficiently small exponential type satisfies
$f^{(mn+r_j)}(s_j)\in{\mathbb Z}$ for $0\le j\le m-1$ and all sufficiently
large $n$, then $f$ is a polynomial. Under suitable assumptions on
$s_0,s_1,\dots,s_{m-1}$ and $r_0,\dots,r_{m-1}$, we introduce interpolation
polynomials $\Lambda_{nj}$, ($n\ge 0$, $0\le j\le m-1$) satisfying $$
\Lambda_{nj}^{(mk+r_\ell)}(s_\ell)=\delta_{j\ell}\delta_{nk},
\quad\hbox{for}\quad n, k\ge 0 \quad\hbox{and}\quad 0\le j, \ell\le m-1 $$ and
we show that any entire function $f$ of sufficiently small exponential type has
a convergent expansion $$ f(z)=\sum_{n\ge 0}
\sum_{j=0}^{m-1}f^{(mn+r_j)}(s_j)\Lambda_{nj}(z). $$ The case $r_j=j$ for $0\le
j\le m-1$ involves successive derivatives $f^{(n)}(w_n)$ of $f$ evaluated at
points of a periodic sequence ${\mathbf{w}}=(w_n)_{n\ge 0}$ of complex numbers,
where $w_{mh+j}=s_j$ ($h\ge 0$, $0\le j\le m$). More generally, given a bounded
(not necessarily periodic) sequence ${\mathbf{w}}=(w_n)_{n\ge 0}$ of complex
numbers, we consider similar interpolation formulae $$ f(z)=\sum_{n\ge
0}f^{(n)}(w_n)\Omega_{{\mathbf{w}},n}(z) $$ involving polynomials
$\Omega_{{\mathbf{w}},n}(z)$ which were introduced by W.~Gontcharoff in 1930.
Under suitable assumptions, we show that the hypothesis
$f^{(n)}(w_n)\in{\mathbb Z}$ for all sufficiently large $n$ implies that $f$ is
a polynomial.
