Probabilistic and combinatorial interpretations of Euler's exp-log recursion

Early algebraic forms of Euler's recursion include the Girard–Newton–Waring identities between coefficients of a polynomial and sums of powers of its roots. As a matter of analysis, the exp-log recursion appears in Euler's calculus text of 1755. With suitable parameters, this recursion generates polynomials known by different names in various contexts: generalized binomial coefficients and Stirling numbers, Newton polynomials, symmetric functions, cycle indices, Bell polynomials, moment polynomials, Faber polynomials, and polynomials of binomial type. Insight into the structure of these polynomials is provided by interpretations of the exp-log relation in various settings, especially its probabilistic interpretations in terms of random permutations, infinitely divisible distributions, and the fluctuation theory of random walks.