Let
${s}_{0},{s}_{1},\dots ,{s}_{m1}$ be complex
numbers and
${r}_{0},\dots ,{r}_{m1}$ rational
integers in the range
$0\le {r}_{j}\le m1$.
Our first goal is to prove that if an entire function
$f$ of sufficiently small
exponential type satisfies
${f}^{\left(mn+{r}_{j}\right)}\left({s}_{j}\right)\in \mathbb{Z}$
for
$0\le j\le m1$ and all sufficiently
large
$n$, then
$f$ is a polynomial. Under
suitable assumptions on
${s}_{0},{s}_{1},\dots ,{s}_{m1}$
and
${r}_{0},\dots ,{r}_{m1}$, we introduce
interpolation polynomials
${\Lambda}_{nj}$
($n\ge 0$,
$0\le j\le m1$)
satisfying
$${\Lambda}_{nj}^{\left(mk+{r}_{\ell}\right)}\left({s}_{\ell}\right)={\delta}_{j\ell}{\delta}_{nk}\phantom{\rule{1em}{0ex}}forn,k\ge 0\phantom{\rule{0.3em}{0ex}}and\phantom{\rule{0.3em}{0ex}}0\le j,\ell \le m1,$$
and we show that any entire function
$f$
of sufficiently small exponential type has a convergent expansion
$$f\left(z\right)=\sum _{n\ge 0}\sum _{j=0}^{m1}{f}^{\left(mn+{r}_{j}\right)}\left({s}_{j}\right){\Lambda}_{nj}\left(z\right).$$
The case
${r}_{j}=j$ for
$0\le j\le m1$ involves successive
derivatives
${f}^{\left(n\right)}\left({w}_{n}\right)$ of
$f$ evaluated at points of
a periodic sequence
$w={\left({w}_{n}\right)}_{n\ge 0}$ 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
$w={\left({w}_{n}\right)}_{n\ge 0}$ of
complex numbers, we consider similar interpolation formulae
$$f\left(z\right)=\sum _{n\ge 0}{f}^{\left(n\right)}\left({w}_{n}\right){\Omega}_{w,n}\left(z\right)$$
involving polynomials
${\Omega}_{w,n}\left(z\right)$
which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the
hypothesis
${f}^{\left(n\right)}\left({w}_{n}\right)\in \mathbb{Z}$ for all
sufficiently large
$n$
implies that
$f$
is a polynomial.
