Some Binomial Sums Involving Absolute Values

Some Binomial Sums Involving Absolute Values

Richard P. Brent
Australian National University
Canberra, ACT 2600
Australia

Hideyuki Ohtsuka
Bunkyo University High School
1191-7, Kami, Ageo-City
Saitama Pref., 362-0001
Japan

Judy-anne H. Osborn
The University of Newcastle
Callaghan, NSW 2308
Australia

Helmut Prodinger
Stellenbosch University
7602 Stellenbosch
South Africa

Abstract:

We consider several families of binomial sum identities whose definition involves the absolute value function. In particular, we consider centered double sums of the form

$\begin{displaymath}S_{\alpha,\beta}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell} \vert k^\alpha-\ell^\alpha\vert^\beta,\end{displaymath}$

obtaining new results in the cases $\alpha = 1, 2$ . We show that there is a close connection between these double sums in the case $\alpha=1$ and the single centered binomial sums considered by Tuenter.

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Received January 30 2016; revised version received March 3 2016. Published in Journal of Integer Sequences, April 6 2016.

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