Zhao -Identities involving reciprocals of binomial coefficients
- B Sury, T Wang
- Journal of Integer Sequences
We obtain expressions for sums of the form ∑m j=0(−1) j ( m j ) ( n+j j ) and deduce, for an even integer d ≥ 0 and m = n > d/2, that this sum is 0 or 1 2 according as to whether d > 0 or not. Further, we prove for even d > 0 that ∑d l=1 cl−1 (−1) ( n l ) l! (l+1) ( 2n l+1 ) = 0 where cr = 1 r! ∑rs=0(−1)s(rs)(r − s + 1)d−1. Similarly, we show when d > 0 is even that ∑d r=0 ar r! ( n r+1 ) ( 2n r+1 ) = 0 where ar = (−1)d+r r! ∑r s=0(−1) (r s ) (r − s + 1)d. Introduction Identities involving binomial coefficients usually arise in situations where counting is carried out in two different ways. For instance, some identities obtained by William Horrace  using probability theory turn out to be special cases of the Chu-Vandermonde identities. Here, we obtain some generalizations of the identities observed by Horrace and give different types of proofs; these, in turn, give rise to some other new identities. In particular, we evaluate sums of the form ∑m j=0(−1)j (mj ) ( j ) and deduce that they vanish when d is even and m = n > d/2. It is well-known  that sums involving binomial coefficients can usually be expressed in terms of the hypergeometric functions but it is more interesting if such a function can be evaluated explicitly at a given argument. Identities such as the ones we prove could perhaps be of some interest due to the explicit evaluation possible. The papers ,  are among many which deal with identities for sums where the binomial coefficients occur in the denominator and we use similar methods here. 1. Horrace’s identities other proofs and generalizations We start with the identities in Horrace’s paper which he deduced using probability theory. 2000 Mathematics Subject Classification. 11B65, 05A19.