Integer-Valued Analytic Functions in a Half-Plane

A classical theorem of Polya states that if f is an entire function taking integer values at the non-negative integers and satisfying the growth condition \(f(z) = o (\mid z \mid^M2^{\mid z \mid}){\rm as}\ z \rightarrow \infty\), for some M > 0, then there exist polynomials P1, P2 with f(z) ≡ P1(z)2z + P2(z). It is shown that the same result holds for functions analytic in a half-plane Re z ≥ A.