i need to show the following identity (LHS = RHS):
Let $\mathbb{N}_0 = \{0,1,2,...\}$ and $n \in \mathbb{N}_0$ and let $d\in\mathbb{N},d\geq 2$.
$\displaystyle \mathrm{(LHS)}:~\sum_{k=0}^d ~\sum_{j_1,\dotsc,j_d \in \mathbb{N}_0 \\ j_1+\dotsc+j_d = n-k} (-1)^k\binom{d}{k}C(F_1(j_1),\dotsc,F_d(j_d)) \\\\ \displaystyle \mathrm{(RHS)}:~\sum_{j_1,\dotsc,j_d \in \mathbb{N}_0 \\ j_1+\dotsc+j_d = n}~\sum_{i_1,\dotsc,i_d \in \{0,1\}} (-1)^{i_1+...+i_d} C(F_1(j_1-i_1),\dotsc,F_d(j_d-i_d)) $
By $C$ i denote a copula and by $F_1,\dotsc,F_d$ i denote univariate distribution functions on $\mathbb{N}_0$.
I tried already to prove the result by induction after $n$, but unfortunately it did not work out.
Thanks in advance and best regards.