I'm trying to prove it in Mujica's book "Complex analysis in Banach spaces" which states the following. I would be grateful if someone could prove it.
Let E and F be Banach Spaces and let $P= P_0+P_1+\ldots +P_m$ where $P_j \in P_a(^jE;F)$ for $j=0\ldots,m$.
Show that the homogeneous polynomial $P_m$ is given by the formula:
$P_m(x)= \dfrac{1}{m!\; 2^m}\sum\limits_{\varepsilon_j=\pm 1} \varepsilon_1 \ldots \varepsilon_m P(x_0+\varepsilon_1x+\ldots+\varepsilon_m x) $
for all $x_0,x\in E$.
Notations: A mapping $P:E \rightarrow F$ is said to be an $m$-homogeneous polynomial if there exists $A\in L_a(^m E;F)$ such that $P(x)=Ax^m$ for every $x\in E$. We shall denote by $P_a(^m E;F) $such that the vector space of all $m$-homogeneous polynomials from $E$ into $F$.