Fred Gross

Prime entire functions

Factorizations of various functions are discussed. Complete factorizations of certain classes of functions are given. In particular it is shown that there exist primes of arbitrary growth.

Further results on prime entire functions

Let H denote the set of all the entire functions f(z) of the form: f(z) = h(z)eM + k(z) where p(z) is a nonconstant polynomial of degree m, and A(# 0), k (# constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinnas theory of meromorphic functions and some elementary facts about algebraic functions.

On the entire solutions of a functional equation in the theory of fluids

In this paper, we shall give necessary and sufficient conditions for the existence of entire solutions to the functional equation ψ2+gφ2=h, where g and h are given nonzero polynomials in z. This functional equation arises when one studies the Percus–Yevick integral equation of hard sphere mixture. The contruction of all such entire solutions ψ and φ is presented. Also, we shall show that it is possible, in some cases, to prove the existence of solutions with prescribed asymptotic properties at z=∞.

Primeable entire functions

An entire function F(z) = f(g(z)) is said to have f(z) and g(z) as left and right factors respe2tively, provided that f(z) is meromorphic and g(z) is entire (g may be meromorphic when f is rational). F(z) is said to be prime (pseudo-prime) if every factorization of the above form implies that one of the functions f and g is bilinear (a rational function). F is said to be E -prime ( E -pseudo prime) if every factorization of the above form into entire factors implies that one of the functions f and g is linear (a polynomial). We recall here that an entire non-periodic function f is prime if and only if it is E -prime [5]. This fact will be useful in the sequel.

A reduction of a type of functional-differential equation

By power series methods it is shown how a functional relation between a set of derivatives of an entire function can sometimes be shown to imply the existence of a much simpler functional relation between the same derivatives of this entire function.

Functional differential equations

It is shown that if an entire function f along with a finite number of its derivatives satisfies a seemingly very weak type of functional equation, then fsatisfies an algebraic differential equation. The proof involves linear algebra and Nevanlinna theory. There are no assumptions concerning the growth of f (the proof given is for transcendental functions f but, clearly, this is the only case in which a proof is needed). Also, there are no assumptions concerning the distribution of values of f.

On periodic entire functions

SummaryIn this paper criteria for deciding on the periodicity of entire functions which have certain forms or satisfy certain functional equations have been provided.