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Written: Feb. 22, 2008.
Appeared in INTEGERS v. 8(2008), A36.
At the last Joint Annual Mathematics meeting, that took place in San Diego
in early Jan., 2008, the great hypergeometric whiz
John Greene gave us an intriging
question, that we are unable to answer (yet). But, as is so often the case,
it inspired new research that is even more important than answering this
or that question. The present research shows that you don't have to be a brute
to use brute force, but indeed rather clever, since brute brute force can't
go very far, but clever brute force sure can!
Added May 6, 2016: See this very interesting
article,
whose author, WENCHANG CHU, kindly allowed us to post here, that proves a conjecture made in our article.
Important: This article is accompanied by Maple
package
BruteTwoFone
that empirically searches for interesting hypergeometric identities,
and immediately proves them (of course).
Sample Input and Output

To search for exact evaluations (up to hypergeometrics of degree 4)
of the form F(an,bn+b0,cn+c0,x) with 1 ≤ a,b,c ≤ 4,
1 ≤ b0,c0 ≤ 2, and b0,c0 integers (i.e. rational
numbers with denominator 1)
the input
will produce the following
output

To search for exact evaluations (up to hypergeometrics of degree 4)
of the form F(an,bn+b0,cn+c0,x) with 1 ≤ a,b,c ≤ 4,
1 ≤ b0,c0 ≤ 2, and b0,c0 are rational
numbers with denominator 2,
the input
will produce the following
output

To search for exact evaluations (up to hypergeometrics of degree 4)
of the form F(an,bn+b0,cn+c0,x) with 1 ≤ a,b,c ≤ 4,
1 ≤ b0,c0 ≤ 2, and b0,c0 are rational
numbers with denominator 3,
the input
will produce the following
output

To find the list of inequivalent exact evaluations, in terms of
the rising factorial,
found above (that excludes those in
the two infinite families described in the paper, and of course,
those implied by classical theorems), using the precomuted output,
the input
will produce the following
output

To find the list of inequivalent exact evaluations, in terms
of the Gamma function,
(that excludes those in
the two infinite families described in the paper, and of course,
those implied by classical theorems), using the precomuted output,
the input
will produce the following
output

To find the closedform evaluations for F(2n,1/2+i,3n1/2+j,3) for
5 ≤ i,j ≤ 5 ,
the input
will produce the following
output
Christian Krattenthaler kindly agreed
to post his nice
sketch of the proof of Theorem 1
Added Feb. 28, 2008:
Ira Gessel
found
yet another proof of Theorem 1,
using the "snakeoil method".
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