Shifting powers in Spivey's bell number formula

Abstract

In this paper, we consider extensions of Spivey's Bell number formula wherein the argument of the polynomial factor is translated by an arbitrary amount. This idea is applied more generally to the r-Whitney numbers of the second kind, denoted by W(n; k), where some new identities are found by means of algebraic and combinatorial arguments. The former makes use of innite series manipulations and Dobinski-like formulas satised by W(n; k), whereas the latter considers distributionsof certain statistics on the underlying enumerated class of set partitions. Further- more, these two  approaches provide new ways in which to deduce the Spivey formula for W(n; k). Finally, we establish an analogous result involving the r-Lah  numbers  wherein the order matters in which the elements are written within the blocks of the aforementioned set partitions.