Evaluation of binomial double sums involving absolute values

TL;DR

This paper derives explicit formulas for complex binomial double sums involving absolute values, expressing them as linear combinations of four basic functions with rational coefficients, using algorithmic and integral methods.

Contribution

It introduces a novel approach to evaluate binomial double sums with absolute values, providing explicit formulas and two different proof techniques.

Findings

Double sums can be expressed as linear combinations of four basic functions.

Explicit formulas involve rational coefficients in terms of n.

Results extend to sums with independent parameters m and n.

Abstract

We show that double sums of the form $$ \sum_{i,j=-n} ^{n} |i^sj^t(i^k-j^k)^\beta| \binom {2n} {n+i} \binom {2n} {n+j} $$ can always be expressed in terms of a linear combination of just four functions, namely $\binom {4n}{2n}$, ${\binom {2n}n}^2$, $4^n\binom {2n}n$, and $16^n$, with coefficients that are rational in $n$. We provide two different proofs: one is algorithmic and uses the second author's computer algebra package Sigma; the second is based on complex contour integrals. In many instances, these results are extended to double sums of the above form where $\binom {2n}{n+j}$ is replaced by $\binom {2m}{m+j}$ with independent parameter $m$.