Exercise 13, § 5

(Divided powers of a series.) The notation is as in Exercises 10 and 11. Let E and F be two k-modules and a a series over E with values in F. Denote by a the homogeneous polynomial law of degree 1 of $\Gamma(E)$ into F defined by a. For any integers p, $m \in \mathbf{N}$ we denote by $\sigma_m^{(p)}$ the series associated with the polynomial law $\gamma_p \circ a \circ (\gamma_0 + \gamma_1 + ... + \gamma_r)$ (cf. Exercise 10). a) Show that as m tends to $\infty$, the sequence $\sigma_m^{(p)}$ converges in $S(E, \Gamma_p(F))$ to a series $\sigma^{(p)}$. This series $\sigma^{(p)}$ is called the divided p-th power of the series a.

b) Show that if $(x_i)_i$, is a family of elements of E and $\eta \in \mathbf{N}^{(1)}$, then

where $\mathscr{L}$ is the set of all mappings with finite support of $\mathbf{N}^{(1)}$ into $\mathbf{N}$ satisfying the conditions $\sum \xi(a) = p$ and $\sum \xi(a) a = \eta$ (Apply Ex. 10, a)).

c) Show that if $a, \tau \in S(E, F)$, then for each $p \in \mathbf{N}$ we have

(product calculated in the algebra $S(E, \Gamma(E))$, cf. Exercise 12).

d) Show that for every integer $p \in \mathbf{N}$ we have $\sigma^p = p! \, \sigma^{(p)}$, where $\sigma^p$ is the p-th power of a in the algebra $S(E, \Gamma(F))$.

e) Let $A_+$ and $A_-$ be the respective coproducts in $\Gamma(E)$ and $\Gamma(F)$. Show that for all $p \in \mathbf{N}$ we have