Christian Krattenthaler and C. Schneider

Evaluation of binomial double sums involving absolute values

(36 pages)

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.

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