Generalized Bell numbers and zeros of successive derivatives of an entire function

Six different formulations equivalent to the statement that, for n ? 2, the sum ∑_{k = 1}ⁿ (?1)^kS(n, k) ≠ 0, where the S(n, k) are Stirling numbers of the second kind, are shown to hold. Using number-theoretic methods, a sufficient condition for the above statement to be true for a set of positive integers n having density 1 is then obtained. It remains open whether it is true for all n > 2. The equivalent statements then yield information on the irreducibility of the polynomials ∑_{k = 1}ⁿS(n, k)t^{k = 1} over the rationals, the nonreal zeros for successive derivatives $\text{(}\text{d}\text{dz}\text{)}{}^{\mathrm{n}}\text{exp}\text{(e}{}^{\mathrm{iz}}\text{)}$, a gap theorem for the nonzero coefficients of exp(?e^z), and the continuous solution of the differential-difference equation $\text{?(x) = 1, 0 ? x < 1, ?′(x) = ?¦x¦?(x ? 1), 1 ? x < ∞}$, where ∥ denotes the greatest integer function.