Abstract

Starting with divided differences of binomial coefficients, a class of multivalued polynomials (three parameters), which includes Bernoulli and Stirling polynomials and various generalizations, is developed. These carry a natural and convenient combinatorial interpretation. Some particular calculations are done and several factorization results are proven and conjectured. 1. Introduction Our study began with a binomial identity involving an alternating sum of vector space dimensions, which arose in the course of proving Bezout's Theorem. The identity led to consideration of a class of polynomials, which are best understood as higher order divided differences of binomial coefficients. These polynomials are closely related to many of the standard polynomials of combinatorial analysis, in particular to the Stirling polynomials, which determine their coefficients. Although it was not possible to find a closed form in general, which was the original intention, calculation of particular va...

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