On the type of a polynomial f, when f*(0)f−f(0)f* is a monomial

Abstract

A polynomial is said to be of type ( p 1 , p 2 , p 3 ) relative to a directed line in the complex plane if, counting multiplicities, it has p 1 zeros to the left of, p 2 zeros on, and p 3 , zeros to the right of the line. In this paper we determine explicitly the types of all polynomials belonging to a very restricted (but infinite) family of polynomials. A polynomial f belongs to this family if and only if its coefficients are such that the polynomial f * (0) f (z)− f (0) f * (z) is a monomial; here f * denotes the reflection of f in the directed line. A special case of the present result appeared in an earlier publication.