by Christian Krattenthaler and Doron Zeilberger
Written March 12, 1997.
When I visited the Institute for Advanced Study, in the Fall Semester of 1993, Enrico Bombieri showed me a family of intriguing conjectured determinant evaluations (and more general statements about the gcd of minors, in case the matrix is rectangular), conjectured by himself in collaboration with David Hunt and Alf van der Poorten. These conjectures would have far-reaching consequences in Diophantine approximation (see ref. [2] of this paper). In this paper, we prove the first case that they can't do (in fact a generalization of it, that they also formulated, but couldn't prove.) Here is the story behind-the-scenes of how the present paper came to be.
Since the simplest case (that they knew how to prove by a different method) is a special case of MacMahon's determinant, that I realized was an immediate consequence of Dodgson's Rule, I was sure that the general case would be also so doable. However, there were some `minor' technical problems, mainly getting 0/0 (perhaps we need a q-analogue?). The other problem was that I needed two extra parameters. I was able to generalize the conjecture by introducing one extra parameter,x, but was unable to find yet another extra parameter, that would set the induction in Rev. Charles's method rolling.
When I distributed the above note to my E-friends, amongst them Christian Krattenthaler, one of the greatest maestros in the world in determinant-evaluations, he asked me for a copy of the Bombieri-Hunt-var der Poorten paper, which I sent him back right away (by E-mail of course.) Then I didn't hear from him for a long time.
When I met Christian in person, this last October 1996, at the MSRI workshop on Enumeration and Posets, and asked him whether he made any progress, he replied that his powerful method is inapplicable, because it requires an extra parameter. I replied that I had this extra parameter, but for MY way of doing determinants I need TWO extras. He got real excited, and said, `for MY (i.e. Christian's) method, one extra parameter suffices', please show it to me. When I came back home, I E-mailed him my conjectured generalization involving the extra x. Then I forgot all about it.
Five months, and fifty-one pages later, Christian did it! Using his brilliant method, and his equally brilliant Mathematica package HYP (that encodes much of what humans did on hypergeometrics in the last 200 years, and has amazing pattern-recognization and transformation features), he was able to finish it up. Most of the identities proved to be too hairy for the (current implementation of) EKHAD, but EKHAD did play a (secondary) role in the creative process (see the introduction.).
The generosity I showed in revealing my extra parameter x to Christian was more than rewarded by the extreme generosity that he demonstrated by making me a co-author. I hope that he is being sincere when he is asserting that without the extra x, he would never have been able to do it. At any rate, I am very honored and pleased to have my Krattenthaler number go down from 2 to 1.
Brace yourself, this is a 51-page-long paper!