Example 2

Let us take for $U$ the composition operator which, to every function $f$ defined on $\mathbf{R}_+$, associates the function $x \mapsto f(x + 1) - f(x)$; then

we have seen (VI, p. 275) that the restriction of $U$ to $\mathbf{C}[X]$ is equal to $e^D - 1$. Since, on the other hand, $u(\lambda) = e^{\lambda} - 1$, the operator $U$ is regular; we shall see (VI, p. 288) how to determine the Bernoulli numbers $b_n$ by calculating an expansion of $\frac{\lambda}{e^{\lambda} - 1}$ as a convergent entire series. On applying formula (27) of VI, p. 278 to a primitive of a function $f$ admitting a continuous $n^{th}$ derivative on $\mathbf{R}_+$, we obtain

with

This lemma is a particular case of a general result that we will prove later; here is the proof. If an entire series $\sum_{n=0}^{\infty} c_n \lambda^n$ is absolutely convergent for $\lambda = \lambda_0$ then for every integer $k \geqslant 0$ the series $\sum_{n=0}^{\infty} c_{n+k} \lambda^n$ is normally convergent for $|\lambda| \leqslant |\lambda_0|$, so is continuous on this disc (Gen. Top., X, p. 283); one concludes that $\sum_{n=k+1}^{\infty} c_n \lambda^n = o(\lambda^k)$ on a neighbourhood of $0$. The lemma then follows from the uniqueness of the coefficients of an asymptotic expansion of a function in terms of the $\lambda^n$ (V, p. 223).

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