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