Coupon collector's probabilities and generating function: analytic   approach
  Despite the coupon collector's problem has simple probabilistic solution using inclusion/exclusion principle \cite{PolyaUrn}, starting from a particular type of recurrence differential equation it is used an analytic approach to recover explicit probabilities and generating function. Note that this kind of recurrence differential equations often appear in Polya Urn context \cite{PolyaUrn}. 
Andrea Monsellato
math.PR