Some Nice Sums Are Almost As Nice If You Turn Them Upside Down

By Moa Apagodu and Doron Zeilberger

Written: July 17, 2009.

[Appeard in Journal of Combinatorics and Number Theory v. 2 (2010), issue 1, paper number 5]

.pdf   .ps   LaTeX source  
[Added Dec. 20, 2011: I now regret publishing this article in this sketchy journal. It is a new expensive print journal, that my University's Library (and, for that matter, any place I checked) does not carry. I was under the false impression that it is a free on-line journal, but I was wrong. Authors must pay a fee for it to become freely on-line. Of course nowadays, most articles are freely available in the arxiv and/or the authors' websites, so the only possible purpose is "prestige", but at least as much prestige can be gotten from FREE on-line journal like Electronic Journal of Combinatorics and INTEGERS]

The sum of all the binomial coefficients n!/(k!(n-k)!) is as nice as can be, namely 2n, but the sum of their reciprocals is also nice (but not quite as nice). Indeed in 1981, Rockett proved that the sum of the reciprocals is (n+1)/2n times Sum(2j/(j+1),j=0..n).

But what about other classics? Like Chu-Vandermonde and Dixon? Here we show that the same phenomenon still holds, but some people may argue that the "almost nice" that we claim is pretty ugly. Well, beauty is in the eyes of the beholder, and at any rate we formally define what we mean by "almost nice", and by that definition, we are safely correct in our assertions.

Checking Maple Program

Important: This article is accompanied by a small Maple program

Sample Input and Output for checkUpsideDown

Doron Zeilberger's List of Papers

Doron Zeilberger's Home Page