Binomial sums and functions of exponential type

Let (a(n))(ngreater than or equal to0) be a sequence of complex numbers, and, for ngreater than or equal to0, let b(n) = Sigma(k=0)(n) ((n)(k))a(k) and cn = Sigma(k=0)(n) ((n)(k))(-1)(n-k)a(k). A number of results are proved relating the growth of the sequences (b(n)) and (c(n)) to that of (a(n)). For example, given p greater than or equal to 0, if b(n) = O(n(p)) and c(n) = O(e(epsilonrootn)) for all epsilon > 0, then a(n) = 0 for all n > p. Also, given 0 < rho < 1, then b(n), c(n) = O(e(epsilonnrho)) for all epsilon > 0 if and only if n(1/rho-1) \a(n)\(1/n) --> 0. It is further shown that, given beta > 1, if b(n), c(n) = O(beta(n)), then a(n) = O(alpha(n)), where alpha = rootbeta(2) - 1, thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredients of the proofs are a Phragmen-Lindelof theorem for entire functions of exponential type zero, and an estimate for the expected value of e(phi)(X), where X is a Poisson random variable.