Vol. 9, No. 4, 2020

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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 s0,s1,,sm1 be complex numbers and r0,,rm1 rational integers in the range 0 rj m 1. Our first goal is to prove that if an entire function f of sufficiently small exponential type satisfies f(mn+rj)(sj) for 0 j m 1 and all sufficiently large n, then f is a polynomial. Under suitable assumptions on s0,s1,,sm1 and r0,,rm1, we introduce interpolation polynomials Λnj (n 0, 0 j m 1) satisfying

Λnj(mk+r)(s ) = δjδnkfor n,k 0 and 0 j, m 1,

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

f(z) = n0 j=0m1f(mn+rj)(s j)Λnj(z).

The case rj = j for 0 j m 1 involves successive derivatives f(n)(wn) of f evaluated at points of a periodic sequence w = (wn)n0 of complex numbers, where wmh+j = sj (h 0, 0 j m). More generally, given a bounded (not necessarily periodic) sequence w = (wn)n0 of complex numbers, we consider similar interpolation formulae

f(z) = n0f(n)(w n)Ωw,n(z)

involving polynomials Ωw,n(z) which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis f(n)(wn) for all sufficiently large n implies that f is a polynomial.

Keywords
Lidstone series, entire functions, transcendental functions, interpolation, exponential type, Laplace transform, method of the kernel
Mathematical Subject Classification
Primary: 30D15
Secondary: 41A58
Milestones
Received: 30 November 2019
Revised: 1 May 2020
Accepted: 15 May 2020
Published: 5 November 2020
Authors
Michel Waldschmidt
Faculté Sciences et Ingénierie
Sorbonne Université
CNRS
Institut Mathématique de Jussieu - Paris Rive Gauche
Paris
France