Permutable entire functions satisfying algebraic differential equations

In this paper we characterize the relation between two entire functions which are permutable and satisfy certain algebraic differential equations.     

Permutable functions

Summary The power functions and the Chebyshev polynomials are examples of families of permutable functions. Recently it was shown how to generalize this idea to polynomials of several variables. In this article the restriction of being polynomials is removed. It is shown how to make a ring and then a field of permutable functions of several variables. The uniqueness problem is discussed.     

Uniqueness of entire functions

Let f be a nonconstant entire function and let a be a meromorphic function satisfying T(r,a)=S(r,f) and a?a′. If f(z)=a(z)⇔f′(z)=a(z) and f(z)=a(z)⇒f″(z)=a(z), then f≡f′, and a?a′ is necessary. This extended a result due to Jank, Mues and Volkmann.     

On entire matrix-valued functions

Let A(z) be a given polynomial n×n matrix with det A(z)=1. It is shown that there exists an entire matrix-valued function X(z) such that expX(z)=A(z) if and only if the eigenvalues of A(z) are independent of z.     

Certain classes of entire functions

Journal d'Analyse Mathématique -