A PROBABILISTIC APPROACH TO SOME OF EULERS NUMBER-THEORETIC IDENTITIES

Probabilistic proofs and interpretations are given for the q-binomial theorem, q-binomial series: two of Euler's fundamental partition ident ities, and for q-analogs of product expansions for the Riemann zeta an d Euler phi functions. The underlying processes involve Bernoulli tria ls with variable probabilities. Also presented are several variations on the classical derangement problem inherent in the distributions con sidered.