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

TL;DR

This paper proves that certain transcendental entire functions with small exponential type, which take integer values at many points in their derivatives, must be polynomials, and introduces new interpolation formulas for such functions.

Contribution

It establishes conditions under which entire functions with integer-valued derivatives at specific points are necessarily polynomials, and develops new interpolation formulas involving derivatives at sequences of points.

Findings

Functions with integer derivatives at certain points are polynomials.

Introduces new interpolation formulas for entire functions.

Shows that integer-valued derivative conditions imply polynomiality.

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 $\Lambda_{nj}$, ($n\ge 0$, $0\le j\le m-1$) 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.