Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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 $n$.  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.
['Christian Krattenthaler', 'Carsten Schneider']
I am writing a research paper on a idea Evaluation of binomial double sums involving absolute values
[{"from": "human", "value": "I am writing a research paper on a idea Evaluation of binomial double sums involving absolute values"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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 $n$.  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."}]