A Eulogy for Jack Good

By Doron Zeilberger

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Written: Dec. 2, 2009.
Irving John ("Jack") Good (9 December 1916 - 5 April 2009) was one of my greatest heroes and influencers. On Oct. 25, 2009, I gave a twenty-three minute talk with the present title, and this article is an extended transcript of that talk. As with all my papers, the "accompanying" Maple package is much more important (mathematically, of course, I also talk about the human side of Jack in this article, and this is even more important than any math). In particular, I taught Jack's brilliant ideas to my computer, and now it can do even better than Jack.

# Sample Input and Output for JACK

• To have the computer guess, and (rigorously prove at the same time!) the recurrence for the Dyson product for n from 2 to 8, the
input
gives the
output.
You don't have to be Jack Good to be able to conjecture that this holds for all n, and then prove it for all n (Good did it with the Lagrange interpolation formula).

• To have the computer, guess, and then prove (ab initio) MacMahon's master theorem for two variables, The
input
gives the
output.
You don't have to be a MacMahon to conjecture the recurrence for an arbitrary number of variables, and then prove it.

• To have the computer guess, and immediately prove, the positive analog of Dyson's conjecture for three variables, the
input
gives the
output.
Note that the computer was unable (in reasonable time) to find the analogous recurrence for four variables, and it is unlikely that a uniform description, like 1-m[1] in the original case, exists.

• Another example that does not seem to generalize nicely to many variables is to find a recurrence for the constant term (or any coefficient for that matter) of
(x12+x1x2+x22)a1(x12+x1x3+x32)a2 (x22+x2x3+x32)a3/(x12a3x22a2x32a1)
The
input
gives the
output.

Doron Zeilberger's List of Papers