Let $z \in \mathbb{C}$. We denote $\Omega_z$ the set of points in the plane with complex affixe $Z$ such that $|Z(Z-2z)| < 1$, and $V_n(z) = U_{n+1}(z,-1)$. Show that there exists a non-empty open disk $\Delta$ with center $O$ included in $\Omega_z$ such that $$\forall Z \in \Delta, \quad \frac{1}{1 - 2zZ + Z^2} = \sum_{n=0}^{+\infty} V_n(z) Z^n = \sum_{p=0}^{+\infty} \left(Z^p(2z - Z)^p\right)$$

Using the results of II.C.6, recover the relation $$V_n(z) = \sum_{j=0}^{\lfloor n/2 \rfloor} \binom{n-j}{j} (2z)^{n-2j} (-1)^j$$

Recall the definition of the Cauchy product of two power series and state the theorem relating to it.