Let be complex numbers and rational integers in the range . Our first goal is to prove that if an entire function of sufficiently small exponential type satisfies for and all sufficiently large , then is a polynomial. Under suitable assumptions on and , we introduce interpolation polynomials (, ) satisfying
and we show that any entire function of sufficiently small exponential type has a convergent expansion
The case for involves successive derivatives of evaluated at points of a periodic sequence of complex numbers, where (, ). More generally, given a bounded (not necessarily periodic) sequence of complex numbers, we consider similar interpolation formulae
involving polynomials which were introduced by W. Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis for all sufficiently large implies that is a polynomial.
"On transcendental entire functions with infinitely many derivatives taking integer values at several points." Mosc. J. Comb. Number Theory 9 (4) 371 - 388, 2020. https://doi.org/10.2140/moscow.2020.9.371