Some Binomial Coefficient Identities of Specific and General Type

Abstract

Underpinned by the falling factorial function, we first give proofs of three specific binomial coefficient identities involving Harmonic numbers which are special cases of a 1996 result due to Kirschenhofer. Using this same tool we then go on to develop some other related identities. In particular, two of them lead to a generalised formula - dealing with binomial coefficient sums of certain type - which is believed by the authors to be a new one.