On entire functions of several variables with derivatives of even order taking integer values

Abstract

We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a polynomial. The proof in the one dimensional case relies on Lidstone expansion of the function. For $n$ variables, we need $n+1$ points, having the property that the differences of $n$ of them with the remaining one give a basis of ${\mathbb{C}}^n$. The proof is by reduction to the one variable situation.

arXiv commentTo appear in Journal of Ramanujan Society of Mathematics and Mathematical Sciences (JRSMAMS), Vol. 9, No. 1, December 2021