By Shalosh B. EKHAD

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Written: Oct. 12, 2004.

Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

According to Roger Howe, mathematics is all about miracles, but mathematicians have a special name for them: `theorems'. But according to my hero Greg Chaitin (and I tend to agree with him), math is just a huge collection of facts, most of them too complicated to be appreciated by humans, and the so-called miracles are just incestous degeneracies and/or contingent coincidences between mathematical quantities. Be that as it may, why not humor humans, and try to find, and prove, facts that they would appreciate and enjoy, and if they want to call them miracles, let them!

In this article, I discovered and proved forty `nice' identities by executing the command BOOK in the Maple package twoFone, written by my beloved human owner, Doron Zeilberger.

Added Jan. 23, 2008: I am very grateful to Moa Apagodu for pointing out a typo (a missing sign in Theorem 30). This has now been corrected. Moa also pointed out that Theorems 29 and 38 are "singular" for positive integers n, so the statement should be preceded by "for all n for which the left side makes sense" (in particular for n

Added Feb. 27, 2014: Akihito Ebisu found many new amazing identities, using a novel new method. See his beautiful article and supporting software.

Note added Feb. 27, 2014: Akihito Ebisu pointed out that Theorem 29 of Ekhad's paper is wrong as stated (i.e. taken literally), since F(-2n, -3n+1, -2n+2, -2) is not well-defined. I am sure that one can make it work, and at any rate it is true `formally', in some sense of `formal' that I am sure exists, but both Shalosh and I are too lazy to explore.

Personal Journal of Ekhad and Zeilberger