Some Questions Concerning Computer-Generated Proofs of a Binomial Double- Sum Identy

for all nonnegative integers n (Problem E3376 of Amer. Math. Monthly 97, March 1990, proposed by R. J. Blodgelt). Up to now not too much is known concerning structure and symbolic manipulation of binomial multiple-sum identities. Thus the interest in a ”computer-generated” proof of (1.1) or, more general, of identities of similar type basically arises from the question whether similar algorithmic tools as those recently developed in the frame of Zeilberger’s approach could be applied. If the answer is positive it is to expect that these methods will stimulate and assist further thorough investigations in this area. In a more adequate frame that is closer to canonical form representation, the problem can be viewed as one concerning manipulation of hypergeometric multiple-sum identities. The great relevance of hypergeometric series to binomial coefficient identities was first pointed out by G. Andrews (1974), and by R. Askey. For further references see also Hayden and Lamagna (1986), Roy (1987), Graham, Knuth and Patashnik (1989), or Koornwinder (1991). Consequently, from hypergeometric theory point-of-view the singlesum case can be considered as well-studied, see for instance the books by Bailey (1935), Slater (1966), or Gasper and Rahman (1990). With respect to algorithmic treatment, recently a break-through has been achieved by D. Zeilberger in the frame of the holonomic systems approach (1990a). His ”fast algorithm” (1990b), which is based on Gosper’s