Short note on the convolution of binomial coefficients

Rui Duarte; António Guedes de Oliveira

paper

Short note on the convolution of binomial coefficients

arXiv:1302.2100

Abstract

We know [Rui Duarte and António Guedes de Oliveira, New developments of an old identity, manuscript arXiv:1203.5424, submitted.] that, for every non-negative integer numbers $n, i, j$ and for every real number $ℓ$ , $i + j = n \sum (i 2 i - ℓ) (j 2 j + ℓ) = i + j = n \sum (i 2 i) (j 2 j),$ which is well-known to be $4^{n}$ . We extend this result by proving that, indeed, $i + j = n \sum (i ai + k - ℓ) (j aj + ℓ) = i + j = n \sum (i ai + k) (j aj)$ for every integer $a$ and for every real $k$ , and present new expressions for this value.

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References in corpus (1)

New developments of an old identity