Sách Elementary Proofs for Convolution Identities of Abel and Hagen–Rothe

However to our knowledge, there does not seem to have appeared really elementary proofs
for these identities in classical combinatorics, even though this has long been desirable.
By utilizing the standard method of series–rearrangement that was systematically used
by Wilf [16], this short paper will present elementary proofs for the convolution identities
of Abel and Hagen–Rothe. It may be unexpected that these proofs are surprisingly
simple, which depend upon the following almost trivial fact that the finite differences of
a polynomial results in zero if the polynomial degree is less than the order of differences.