On transcendental entire functions with infinitely many derivatives taking integer values at several points

Let [math] be complex numbers and [math] rational integers in the range [math] . Our first goal is to prove that if an entire function [math] of sufficiently small exponential type satisfies [math] for [math] and all sufficiently large [math] , then [math] is a polynomial. Under suitable assumptions on [math] and [math] , we introduce interpolation polynomials [math] ( [math] , [math] ) satisfying¶Λ(+ℓ)(ℓ)=δℓδ<mspace class="quad" width="1em"/>,≥00≤,ℓ≤−1,¶ and we show that any entire function [math] of sufficiently small exponential type has a convergent expansion¶()=<munder class="msub"> ∑≥0</munder><munderover accent="false" accentunder="false"> ∑=0−1</munderover>(+)()Λ().¶ The case [math] for [math] involves successive derivatives [math] of [math] evaluated at points of a periodic sequence [math] of complex numbers, where [math] ( [math] , [math] ). More generally, given a bounded (not necessarily periodic) sequence [math] of complex numbers, we consider similar interpolation formulae¶()=<munder class="msub"> ∑≥0</munder>()()Ω,()¶ involving polynomials [math] which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis [math] for all sufficiently large [math] implies that [math] is a polynomial.