A Multiple Integral Explicit Evaluation Inspired by The Multi-WZ Method

We give an identity which is conjectured and proved by using an implementation [3] in multi-WZ [5]. 0. Introduction There are relatively few known non-trivial evaluations of n-dimensional integrals, with arbitrary n. Celebrated examples are the Selberg and the Metha-Dyson integrals, as well as the Macdonald constant term ex-conjectures for the various infinite families of root systems. They are all very important. See [1] for a superb exposition of the various known proofs and of numerous intriguing applications. At present, the (continuous version of the) WZ method[5] is capable of mechanically proving these identities only for a fixed n. In principle for any fixed n (even, say, n = 100000), but in practice only for n ≤ 5. However, by interfacing a human to the computer-generated output, the human may discern a pattern, and generalize the computer-generated proofs for n = 1, 2, 3, 4 to an arbitrary n. Using this strategy, Wilf and Zeilberger[5] gave a WZ-style proof of Selberg’s integral evaluation. But just giving yet another proof of an already known identity, especially one that already had (at least) three beautiful proofs (Selberg’s, Aomoto’s, and Anderson’s, see [1]), is not very exciting. In this article we present a new multi-integral evaluation, that was first found using the author’s implementation of the continuous multi-WZ method[3]. Both the conjecturing part, and the proving part, were done by a close human-machine collaboration. Our proof hence may be termed computer-assisted but not yet computer-generated. Department of Mathematics, Temple University, Philadelphia, PA 19104. [email protected]