Proposition 6

Let $U = D^p V = u(D)$ be a composition operator of order $p$ on $K[X]$, $u(S)$ being a formal power series of order $p$ in $K[[S]]$. Then

n being the Appell polynomial of index n attached to U.

By the scholium to th. 1 (VI, p. 271), to establish (15) it is enough to show that for every polynomial $f(Y) \in K[Y]$ one has

Now the first term in (17) is $U_X(f(X + Y))$, and, since $U = u(D)$, the second term in (17) is $U_Y(f(X + Y))$, so the identity (17) reduces to (1) (VI, p. 270).

It now suffices to apply (15) to the composition operator $V^{-1} = D^p / u(D)$ to obtain (16), since, by definition, one has

Note that (16) can also be obtained by multiplying the formal series $S^p / u(S)$ and $\exp(XS)$, taking account of (6).

One says that the formal series (16) is the generating series of the Appell polynomials attached to $U$.