Definition 3

One says that an operator U on E is a composition operator if, for every $a \geqslant 0$, it is permutable with the translation operator by a.

In the notation introduced above, this definition becomes the identity

in $x$ and $a$ ($x \in I, a \geqslant 0$). One can exchange the rôles of $x$ and $a$ in this identity if $x \geqslant 0$, then put $a = 0$; one thus obtains for $x \geqslant 0$

where $U_0$ is the linear form on E which to each function $f \in E$ associates the value $g(0)$ of $g = U(f)$.

If $f$ is a polynomial, one has $f(\xi + x) = \sum_{k=0}^{\infty} \frac{1}{k!} f^{(k)}(\xi) x^k$, and formula (26) shows that $U(f)$ is also a polynomial; restricted to the set of polynomials in $x$, with coefficients in $\mathbf{C}$, (a set which one can identify with the algebra $\mathbf{C}[X]$), the operator $U$ is then a composition operator in the sense of def. I of VI, p. 270, and all the results of the preceding sections can be applied to it.

We again write $u_n$ for the Appell polynomials attached to the operator $U$. To the generalized Taylor expansion of a polynomial (VI, p. 274, formula (13)) there corresponds, for more general functions, the following result: