2019/11/30 by Michel Waldschmidt, Waldschmidt, Michel
Mathematics · #30D15 #41A58 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Number Theory (math.NT) #math.NT #msc:30D15 #msc:41A58
paper · pdf · doi:10.48550/arxiv.1912.00173
28 pages
arxiv created 2019/11/30 · openalex publication_date 2019/11/30 · arxiv updated 2019/12/03 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
Given a subset S=\s0, s1\ of the complex plane with two points and an infinite subset \mathscr S of S× \mathbb N, where \mathbb N=\0,1,2,…\ is the set of nonnegative integers, we ask for a lower bound for the order of growth of a transcendental entire function f such that f(n)(s)∈\mathbb Z for all (s,n)∈\mathscr S. We first take \mathscr S=\s0,s1\× 2\mathbb N, where 2\mathbb N=\0,2,4,…\ is the set of nonnegative even integers. We prove that an entire function f of sufficiently small exponential type such that f(2n)(s0)∈\mathbb Z and f(2n)( s1)∈\mathbb Z for all sufficiently large n must be a polynomial. The estimate we reach is optimal, as we show by constructing a noncountable set of examples. The main tool, both for the proof of the estimate and for the construction of examples, is Lidstone polynomials. Our second example is (\s0\× (2\mathbb N+1))∪( \ s1\× 2\mathbb N) (odd derivatives at s0 and even derivatives at s1). We use analogs of Lidstone polynomials which have been introduced by J.M.~Whittaker and studied by I.J.~Schoenberg. Finally, using results of W.~Gontcharoff, A. J.~Macintyre and J.M.~Whittaker, we prove lower bounds for the exponential type of a transcendental entire function f such that, for each sufficiently large n, one at least of the two numbers f(n)(s0), f(n)(s1) is in \mathbb Z.