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

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 $f^{(mn+r_j)}(s_j)\in{\mathbb Z}$ 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 $Λ_{nj}$, ($n\ge 0$, $0\le j\le m-1$) satisfying $$ Λ_{nj}^{(mk+r_\ell)}(s_\ell)=δ_{j\ell}δ_{nk}, \quad\hbox{for}\quad n, k\ge 0 \quad\hbox{and}\quad 0\le j, \ell\le m-1 $$ and we show that any entire function $f$ of sufficiently small exponential type has a convergent expansion $$ f(z)=\sum_{n\ge 0} \sum_{j=0}^{m-1}f^{(mn+r_j)}(s_j)Λ_{nj}(z). $$ 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 $$ f(z)=\sum_{n\ge 0}f^{(n)}(w_n)Ω_{\mathbf{w},n}(z) $$ involving polynomials $Ω_{\mathbf{w},n}(z)$ 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.