Evaluation of binomial double sums involving absolute values

Abstract

We show that double sums of the form $$\sum _{i,j=-n}^n|i^s{j}^t(i^k-j^k{)}^{\beta }|(\genfrac{}{}{0ex}{}{2n}{n+i}\left)\right(\genfrac{}{}{0ex}{}{2n}{n+j})$$ can always be expressed in terms of a linear combination of just four functions, namely $\left(\genfrac{}{}{0ex}{}{4n}{2n}\right)$, ${\left(\genfrac{}{}{0ex}{}{2n}{n}\right)}^2$, $4^n\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$, 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 $\left(\genfrac{}{}{0ex}{}{2n}{n+j}\right)$ is replaced by $\left(\genfrac{}{}{0ex}{}{2m}{m+j}\right)$ with independent parameter $m$.

AmS-LaTeX, 42 pages; final version