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

A polynomial is said to be of type (p₁, p₂, p₃) relative to a directed line in the complex plane if, counting multiplicities, it has p₁ zeros to the left of, p₂ zeros on, and p₃ 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 ƒ belongs to this family if and only if its coefficients are such that the polynomial ƒ^*(0)ƒ(z)−ƒ(0) ƒ^*(z) is a monomial; here ƒ^* denotes the reflection of ƒ in the directed line.

A special case of the present result appeared in an earlier publication.