On transcendental entire functions with infinitely many derivatives taking integer values at several points. (English) Zbl 1451.30059

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.


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)
Full Text: DOI arXiv


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