  We consider a class of generalized binomials emerging in fractional calculus.
After establishing some general properties, we focus on a particular yet
relevant case, for which we provide several ready-for-use combinatorial
identities, including an adapted version of the Pascal's rule. We then
investigate the associated generating functions, for which we establish a
recursive, combinatorial and integral formulation. From this, we derive an
asymptotic version of the Binomial Theorem. A combinatorial and asymptotic
analysis of some finite sums completes the paper.
