Short note on the convolution of binomial coefficients

Abstract

We know [Rui Duarte and Ant\'onio 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 $\ell$, $$ \sum_{i+j=n} \binom{2i-\ell}{i} \binom{2j+\ell}{j} = \sum_{i+j=n}\binom{2i}{i} \binom{2j}{j}, $$ which is well-known to be $4^n$. We extend this result by proving that, indeed, $$ \sum_{i+j=n} \binom{ai+k-\ell}{i} \binom{aj+\ell}{j} = \sum_{i+j=n} \binom{ai+k}{i} \binom{aj}{j} $$ for every integer $a$ and for every real $k$, and present new expressions for this value.