Limits of sums for binomial and Eulerian numbers and their associated distributions

We provide a probabilistic approach using renewal theory to derive some novel identities involving Eulerian numbers and uniform B-splines. The renewal perspective leads to a unified treatment for the normalized binomial coefficients and the normalized Eulerian numbers when studying their limits of sums, as well as their associated distributions — the binomial distributions (Bernstein polynomials) and the Irwin–Hall distributions (uniform B-splines). We further extend the probabilistic perspective to h-Bernstein polynomials (Pólya–Eggenberger distributions) through conditional renewal processes, and derive new limits of various ways of summing for the two special numbers and associated distributions. The proposed probabilistic unification dramatically simplifies the proofs of some identities, which are far from obvious (such as for h-Bernstein polynomials) or do not otherwise even appear promising (such as for Eulerian numbers).