#### Vol. 9, No. 4, 2020

 Recent Issues Volume 10, Issue 1 Volume 9, Issue 4 Volume 9, Issue 3 Volume 9, Issue 2 Volume 9, Issue 1 Volume 8, Issue 4 Volume 8, Issue 3 Volume 8, Issue 2 Volume 8, Issue 1
 The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement founded and published with the scientific support and advice of the Moscow Institute of Physics and Technology ISSN (electronic): 2640-7361 ISSN (print): 2220-5438 Previously Published Author Index To Appear Other MSP Journals
On transcendental entire functions with infinitely many derivatives taking integer values at several points

### Michel Waldschmidt

Vol. 9 (2020), No. 4, 371–388
##### 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 ${f}^{\left(mn+{r}_{j}\right)}\left({s}_{j}\right)\in ℤ$ 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

${\Lambda }_{nj}^{\left(mk+{r}_{\ell }\right)}\left({s}_{\ell }\right)={\delta }_{j\ell }{\delta }_{nk}\phantom{\rule{1em}{0ex}}forn,k\ge 0\phantom{\rule{0.3em}{0ex}}and\phantom{\rule{0.3em}{0ex}}0\le j,\ell \le m-1,$

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

$f\left(z\right)=\sum _{n\ge 0}\sum _{j=0}^{m-1}{f}^{\left(mn+{r}_{j}\right)}\left({s}_{j}\right){\Lambda }_{nj}\left(z\right).$

The case ${r}_{j}=j$ for $0\le j\le m-1$ 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

$f\left(z\right)=\sum _{n\ge 0}{f}^{\left(n\right)}\left({w}_{n}\right){\Omega }_{w,n}\left(z\right)$

involving polynomials ${\Omega }_{w,n}\left(z\right)$ which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis ${f}^{\left(n\right)}\left({w}_{n}\right)\in ℤ$ for all sufficiently large $n$ implies that $f$ is a polynomial.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/moscow

We have not been able to recognize your IP address 3.239.109.55 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.