By Shalosh B. EKHAD
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.
.pdf
.ps
.tex
Written: Oct. 12, 2004.
Exclusively published in the
Personal Journal of Shalosh B. Ekhad and 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 negative integers).
Personal Journal of Ekhad and Zeilberger