A special class of Bell polynomials

Abstract:

We examine the integers $V(n,k)$ defined by means of \[ k!\sum \limits _{n = 0}^\infty {V(n,k){x^n}/n! = {{[x({e^x} + 1) - 2({e^x} - 1)]}^k},} \] and, in particular, we show how these integers are related to the Bernoulli, Genocchi and van der Pol numbers, and the numbers generated by the reciprocal of ${e^x} - x - 1$. We prove that the $V(n,k)$ are also related to the numbers $W(n,k)$ defined by \[ k!\sum \limits _{n = 0}^\infty {W(n,k){x^n}/n! = {{[(x - 2)({e^x} - 1)]}^k}} \] in much the same way the associated Stirling numbers are related to the Stirling numbers. Finally, we examine, more generally, the Bell polynomials ${B_{n,k}}({a_1},{a_2},3 - \alpha ,4 - \alpha ,5 - \alpha , \ldots )$ and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.