Identities for Bernoulli numbers

Question

I arrived at this formula by inductive reasoning, but I don’t know how to prove it.

For any natural numbers $m$ and $k=0,1,2,\ldots, m-1$, $B_i$ - Bernoulli numbers we have: $$\sum_{i=0}^k (-1)^{k-i}\cdot B_i\cdot\binom{m+i-k-1}{i}\cdot\binom{m}{m+i-k-1}=(-1)^{k}\cdot m\cdot\binom{m-1}{k}$$

Answer 1

This identity can be written in the following form $$\sum_{i=0}^k (-1)^{k-i+1} B_i\cdot\binom{k+1}{i}=(-1)^{k+1}(k+1).$$ The latter is equivalent to $B_{k+1}(0)-B_{k+1}(-1)=(k+1)(-1)^k$ which is one of the basic properties of Bernoulli polynomials.