Searching For Strange Hypergeometric Identities By Sheer Brute Force

By Moa Apagodu and Doron Zeilberger

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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 pre-comuted 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 pre-comuted output,
    the input will produce the following output
  • To find the closed-form evaluations for F(-2n,-1/2+i,-3n-1/2+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 "snake-oil method".
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