Towards a Wz Evolution of the Mehta Integral * Doron Zeilberger ?

The celebrated Mehta integral is shown to be equivalent to a simple algebraic-differential identity, which is completely routine for any fixed number of variables. Askey [As] proposed the problem of proving the Mehta (see [M]) integral identity (Mehta) 2 1 exp(-x/2 xnl2 H (xi-xj)2Cdxl""dan-H (cj)! without using Selberg's integral (see [M]). This problem was solved by Anderson [An]. Here we use the method of [WZ1] and[WZ2] to initiate another Selberg-free proof that we believe is of independent interest. We show that (Mehta) for any given n is equivalent to the following elegant identity (d := n(n-1)/2): (Mehta') 0 2 1 0 0 li<jSn 15i<jn H ls<rn (Mehta) is purely routine for any specific n, but at the time of writing we are unable to prove it directly for general n. Of course, we do have a proof, since we e going to show that (Mehta) and (Mehta) e equivalent, but what we are aer is a direct proof. The author is offering a prize of 25 US dollars for such a proof. The present method also obviously extends to the Macdonald-Mehta inteal [M], which w proved by neckner d Regev [BR] for the clsical root systems (see [M]), by Gvan [G] for the exceptional root system Fa, and by Opdam [O] for E6, ET, and Es. It follows that the present approh should also yield new proofs for all the exceptional root systems, at let in principle, but most likely in practice also. More important, it seems to have a high chance of producing a uniform, intrinsic, clsification-independent proof. We leave to the reader, an instructive exercise, the tk of finding the root-system analog of (Mehta) that is equivalent to the now-proved Mehta-Macdonald conjecture, and we offer an additional 25 dollars for an intrinsic proof. Our proposed proof of (Mehta) will be a de,ration rather than a vefication, and will follow the method of [WZ2]. Let us call the len of (Mehta) i(c), and the inteand F(c; x,..., xn). We know from the general theory of [WZ2] that for some r