The variance function of the natural exponential family generated by a measure on n integers

We show a method for computing the variance function of the natural exponential family generated either by a uniform measure on n integers or by a general measure on four values. An expression, involving roots of polynomials, of the variance function of a natural exponential family generated by the measure on n integers is provided. Moreover we show that there exists a natural exponential family which has a nonquadratic variance and satisfies the property of natural exponential family with quadratic variance function stated by Jorgensen, Seshadri and Letac (1989).