Waldschmidt, Michel On transcendental entire functions with infinitely many derivatives taking integer values at several points. (English) Zbl 1451.30059 Mosc. J. Comb. Number Theory 9, No. 4, 371-388 (2020). Summary: 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 \[\Lambda_{nj}^{(mk+r_\ell)}(s_\ell)=\delta_{j\ell}\delta_{nk} \quad\text{for } n, k\ge 0 \text{ and } 0\le j, \ell\le m-1,\] 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. MSC: 30D15 Special classes of entire functions of one complex variable and growth estimates 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) Keywords:entire functions of exponential type; interpolation PDFBibTeX XMLCite \textit{M. Waldschmidt}, Mosc. J. Comb. Number Theory 9, No. 4, 371--388 (2020; Zbl 1451.30059) Full Text: DOI arXiv References: [1] ; Boas, Polynomial expansions of analytic functions. Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F., 19 (1964) · Zbl 0116.28105 [2] 10.2307/1990503 · Zbl 0033.36401 [3] 10.2307/2032936 · Zbl 0068.05502 [4] 10.24033/asens.798 · JFM 56.0260.01 [5] ; Halphén, Bull. Soc. Math. France, 10, 67 (1882) · JFM 14.0367.02 [6] 10.2307/1990741 · Zbl 0058.29702 [7] 10.4213/sm629 [8] 10.2307/1989543 · Zbl 0004.34307 [9] 10.1112/plms/s2-36.1.451 · Zbl 0008.16901 [10] ; Whittaker, Interpolatory function theory. Cambridge Tracts in Mathematics and Mathematical Physics, 33 (1964) · Zbl 0012.15503 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.