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.
Categories: math.NT