On Differential Rings of Entire Functions

Abstract

Consider an entire function / which is a solution of the differential equation \c0(z) + cx(z)D + ... + cm(z)Dm\(fn) = P(f, f./(fc))where c¡(z) are entire functions in a differential ring R and P is a polynomial in a differential field related to R. We prove the following THEOREM.If f satisfies the equation above then f is of finite type in case R = C and of finite exponential order in case R = C[z).We use this result to prove a conjecture made in [2] that entire functions of order p < s, all of whose derivatives at i points are integers in an imaginary quadratic number field, must be solutions of linear differential equations with constant coefficients and therefore of order < 1.

Locations

• Transactions of the American Mathematical Society - View - PDF