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2020 On transcendental entire functions with infinitely many derivatives taking integer values at several points
Michel Waldschmidt
Mosc. J. Comb. Number Theory 9(4): 371-388 (2020). DOI: 10.2140/moscow.2020.9.371

## 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 ${\mathrm{\Lambda }}_{nj}$ ($n\ge 0$, $0\le j\le m-1$) satisfying

${\mathrm{\Lambda }}_{nj}^{\left(mk+{r}_{\ell }\right)}\left({s}_{\ell }\right)={\delta }_{j\ell }{\delta }_{nk}\phantom{\rule{1em}{0ex}}forn,k\ge 0and0\le j,\ell \le m-1,$

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

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

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

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

involving polynomials ${\mathrm{\Omega }}_{w,n}\left(z\right)$ 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.

## Citation

Michel Waldschmidt. "On transcendental entire functions with infinitely many derivatives taking integer values at several points." Mosc. J. Comb. Number Theory 9 (4) 371 - 388, 2020. https://doi.org/10.2140/moscow.2020.9.371

## Information

Received: 30 November 2019; Revised: 1 May 2020; Accepted: 15 May 2020; Published: 2020
First available in Project Euclid: 12 November 2020

zbMATH: 07272356
MathSciNet: MR4170703
Digital Object Identifier: 10.2140/moscow.2020.9.371

Subjects:
Primary: 30D15
Secondary: 41A58

Keywords: entire functions , exponential type , interpolation , Laplace transform , Lidstone series , method of the kernel , transcendental functions  