EVEN COMPOSITIONS OF ENTIRE-FUNCTIONS AND RELATED MATTERS

We examine when the composition of two entire functions f and g is eve n, and extend some of our results to cyclic compositions in general. I f p is a polynomial, then we prove that f o p is even for a non-consta nt entire function f if and only if p is even, odd plus a constant, or a quadratic polynomial composed with an odd polynomial. Similar resul ts are proven for odd compositions. We also show that p o f can be eve n when f and no derivative of f are even or odd, where p is a polynomi al. 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.