Now let $\alpha$ be a pure word with $n_j$ symbols of degree $d_j$. It is easy to prove inductively that:
$n_0 + n_1 + n_2 + ... = 1 + 0.n_0 + 1.n_1 + 2.n_2 + ...,$
i.e. $n_0 = 1 + n_2 + 2n_3 + ...$
I think there is a small mistake here, since $n_j$ should be considered as the number of symbols of degree $j$ and not of degree $d_j$ in $\alpha$.
$n_0 + n_1 +...