91.13 A new proof of a curious identity

Abstract

We give a short WZ-proof of a binomial coefficient identity due to Zhi-Wei Sun. In [3], [2], and [1] the identity m ∑ i=0 (x+m+ 1)(−1) ( x+ y + i m− i )( y + 2i i ) − m ∑ i=0 ( x+ i m− i ) (−4) = (x−m) ( x m ) (1) was proved using generating functions, double recursions and the concept of Riordan Arrays respectively. Here we use the WZ-method to give yet another proof. First we divide both sides of (1) by (x + m + 1) and then try to write the second indefinite sum on the LHS of (1) without the running index m under the summation sign. This is automated in EKHAD in the procedure zeillim and can be run by the command: zeillim(SUMMAND , i,m,M, 0, 0). But we include here the mathematics behind zeillim for the sake of clarity. Consider the indefinite sum