Some binomial sums involving absolute values

TL;DR

This paper investigates binomial sum identities involving absolute values, deriving new formulas for specific cases and revealing connections to existing binomial sums, thus advancing combinatorial sum theory.

Contribution

It introduces new binomial sum identities involving absolute values and links them to known sums, expanding the understanding of such combinatorial expressions.

Findings

Derived new formulas for binomial sums with absolute values for α=1,2

Established connections between double sums and single binomial sums

Extended the theory of binomial sum identities involving absolute values

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 \[S_{\alpha,\beta}(n) := \sum_{k,\;\ell}\binom{2n}{n+k}\binom{2n}{n+\ell} |k^\alpha-\ell^\alpha|^\beta,\] 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.