Summary: Recently, Spivey discovered a novel formula for $B(n+m)$, where $B(n+m)$ is the $(n+m)^{th}$ Bell number. His proof was combinatorial in nature. This paper provides a generating function proof of Spivey's result. It also uses Spivey's formula to determine a new formula for $B(n)$. The paper concludes by extending all three identities to ordinary single variable Bell polynomials.

11B73

Bell number, Stirling number, Bell polynomial (Concerned with sequence )