A linear binomial recurrence and the bell numbers and polynomials

Abstract

Let B(n) denote the nth BELL number. It is well known that B(n) obeys the recurrence relation The goal of this paper is to study arbitrary functions f(n) that obey (0.1), namely, By iterating (0.2), f(n + r) can be written as a linear combination of binomial coefficients with polynomial coefficients Ajr (n), 0 ≤ j ≥ r - 1. The polynomials Ajr (n) have various interesting properties. This paper provides a sampling of these properties, including two new ways to represent B(n) in terms of Ajr (n).