Abstract

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 $f^{\left(mn+r_j\right)}\left(s_j\right)\in \mathbb{Z}$ 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

and we show that any entire function $f$ of sufficiently small exponential type has a convergent expansion

The case $r_j=j$ for $0\le j\le m-1$ 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

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.

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