Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: A Multiple Integral Evaluation Inspired by the Multi-WZ Method. | A Multiple Integral Evaluation Inspired by the Multi-WZ Method Akalu Tefera Department of Mathematics Temple University Philadelphia PA 19122. akalu@ Abstract We give an integral identity which was conjectured and proved by using the continuous version of the multi-WZ method. Submitted May 25 1999 Accepted October 20 1999. Mathematical Reviews Subject Numbers 05 33. 0. Introduction There are relatively few known non-trivial evaluations of k-dimensional integrals with arbitrary k. Celebrated examples are the Selberg and the Mehta-Dyson integrals as well as the Macdonald constant term ex-conjectures for the various root systems. They are all very important. See AAR98 for a superb exposition of the various known proofs and of numerous intriguing applications. At present the continuous version of the WZ method WZ92 is capable of mechanically proving these identities only for a fixed k. In principle for any fixed k even say k 100000 but in practice only for k 5. However by interfacing a human This work will appear in the author s . thesis. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 6 1999 N2 2 to the computer-generated output the human may discern a pattern and generalize the computer-generated proofs for k 1 2 3 4 to an arbitrary k. Using this strategy Wilf and Zeilberger WZ92 gave a WZ-style proof of Selberg s integral evaluation. In this article we present a new multi-integral evaluation that was first found by using the author s implementation of the continuous multi-WZ method which is called SMint1. 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. Now that the result is known and proved it may be of interest to have a non-WZ proof possibly by performing an appropriate change of variables converting the multiintegral to a double integral. My advisor Doron Zeilberger is offering 100 for such a proof provided it does not .
