Even compositions of entire functions and related matters

Abstract

AbstractWe examine when the composition of two entire functions f and g is even, and extend some of our results to cyclic compositions in general. If p is a polynomial, then we prove that f ^ p is even for a non-constant entire function f if and only if p is even, odd plus a constant, or a quadratic polynomial composed with an odd polynomial. Similar results are proven for odd compositions. We also show that p ^ f can be even when f and no derivative of f are even or odd, where p is a polynomial. We extend some results of an earlier paper to cyclic compositions of polynomials. We also show that our results do not extend in general to rational functions or polynomials in two variables.