In a recent question that I am still working on, I needed to consider some sums involving Bernoulli numbers. After a lot of computations and simplifications, I ended up having to manipulate the following sum : $$\sum_{k=\ell+2}^{2\delta}\binom{k}{\ell+2}\binom{2\delta+2}{k+2}B_{2\delta-k}=\sum_{k=0}^{2\delta-2-\ell}\binom{2\delta-k}{\ell+2}\binom{2\delta+2}{k}B_k,$$ where $0\leqslant\ell\leqslant2\delta-2$ and $\delta\geqslant2$. I tried several things, one of which being making use of Weisstein formulas : $$\sum_{k=0}^m\binom{m+1}{k}B_k=\delta_{m,0}.$$
I tried developing the binomial coefficients by making use of the factorial, using the various identities involving them, or even trying summation by parts. Sadly, nothing of the sort worked. I'd also try using the generating function for the sequence $(B_k)_{k\in\mathbb{N}}$, but I have no clue about how I could proceed.
Is there a way to simplify the expression I gave above ?
I'd need something slightly more suitable for algebraic reasoning, but I'm starting to doubt whether that's feasible... Does anyone have any clue/hint/idea on how to manipulate this expression ?
Side note : I plugged it into Mathematica, and it wasn't able to compute anything. But I doubt it has the sufficient knownledge (for instance, Weisstein or Saalschütz).
Update : I have been advised to giving a try to Volkenborn integration, from which the Weisstein formula I gave above is a direct consequence. Therefore, manipulation of an expression such as $$\sum_{k=0}^{2\delta-2-\ell}\binom{2\delta-k}{\ell+2}\binom{2\delta+2}{k}X^k$$
may be useful. Unfortunately, I see no direct computation such as the Binomial expansion that can be used here, and Mathematica only rewrote the expression as : $$\sum_{k=0}^{2\delta-2-\ell}\binom{2\delta-k}{\ell+2}\binom{2\delta+2}{k}X^k=\binom{2\delta}{\ell+2}{_2F_1}(-2\delta-2,2-2\delta+\ell;-2\delta;X).$$
This may be a new starting point though, therefore I added it to this question.
