Generalized Bell numbers and zeros of successive derivatives of an entire function (Q790986)

Six different formulations equivalent to the statement that, for \(n\geq 2\), the sum \(\sum^{n}_{k=1}(-1)^ kS(n,k)\neq 0,\) where the S(n,k) are Stirling numbers of the second kind, are shown to hold. The equivalent statements yield information of the irreducibility of the polynomials \(\sum^{n}_{k=1}S(n,k)t^{k-1}\) over the rationals, the general zeros for successive derivatives \((d/dz)^ n \exp(e^{iz}),\) a gap theorem for the nonzero coefficients of \(\exp(-e^ z),\) and the continuous solution of the differential-difference equation \(f(x)=1,\quad 0\leq x<1,\quad f'(x)=-[x]f(x-1), 1\leq x<\infty\), where [ ] denotes the greatest integer function. Using number-theoretic methods the following sufficient condition is established: Let \(A_ n\) be the sequence defined by \(\exp(1-e^ z)=\sum^{\infty}_{n=0}A_ n z^ n/n!,\) and let z(N) denote the number of zeros \(A_ n\) for \(z=1,2,...,N\). If there exist arbitrarily large primes p for which the minimal period of \(A_ n(mod p)\quad is\quad 2(p^ p-1)/(p-1),\) then \(\lim_{n\to \infty}z(N)/N=0.\) The polynomials \(S_ n(t)=\sum^{n}_{k=1}S(n,k)t^ k\) are of interest, and are shown to have the following property: The set of all zeros of the polynomials \(S_ n(t)\), \(n=1,2,...\), is dense on (- \(\infty,0]\).