A probabilistically attained set of polynomials that generate stirling numbers of the second kind

Using probabilistic arguments, we derive a sequence of polynomials in one variable which generate the Stirling numbers of the second kind. Specifically, S^m"c=(c!/m!)P"c"-"m(c), where S^m"c is the desired Stirling number and P"c"-"m(.) is the polynomial of degree c-m.