On polynomial variance functions

Summary Let be a natural exponential family on and (V, ) be its variance function. Here, is the mean domain of andV, defined on , is the variance of . A problem of increasing interest in the literature is the following: Given an open interval and a functionV defined on , is the pair (V, ) a variance function of some natural exponential family? Here, we consider the case whereV is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, ) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.