RECURRENCE AND DIRECT FORMULAS FOR THE HAL & LAH NUMBERS

Abstract

In this paper we study some properties concerning the Hal )H( k n and Lah numbers ),L( k n which are those that arise in the theory of combinatorics in association with the “upward and downward factorial polynomials” [x]n and [x]n. By using generating function techniques and simple binomial expansions we establish some nive results for the Hal number ),H( k n like the recurrence formula, the generating funcion and the direct (closed) formula, among others properties. Finally we compare these results with the corresponding formulas for the Lah numbers ).L( k n We establish the corresponding recurrence formulas, the generating functions and the closed formulas for this family of numbers, among others properties.