I'm looking for a simplification to the following expression
$$ \sum_{j=0}^m\sum_{l=0}^m \binom{m}{j} \binom{m}{l} h(j+l) $$ where $h$ is a given function.
I looked for a formula of the product of binomial coefficients but I couldn't find it also I tried to simplify it using maple but in vain ! is it complicated this much ?! because it looks like a familiar formula. Any hint or help will be so appreciated :) thank you for your time.
$$\sum_{j=0}^{m}\sum_{l=0}^{m}\binom{m}{j}\binom{m}{l}h(j+l) = \sum_{k=0}^{2m}h(k)\sum_{j+l=k}\binom{m}{j}\binom{m}{l} = \sum_{k=0}^{2m}\binom{2m}{k}h(k)$$ by Vandermonde's identity.