Definition 2

Tag 08HZ, permanent. Cite it as [Bourbaki, Tag 08HZ].

Functions of a Real Variable, chapter VI, § 1. GENERALIZED TAYLOR EXPANSIONS

Let $U = D^p V$ be a composition operator of order $p$ on $\mathbf{K}[X]$ . The polynomial $u_n(X) = V^{-1}(X^n)$ is called the Appell polynomial of index $n$ attached to the operator $U$ .

If $V^{-1} = \sum_{k=0}^{\infty} \frac{1}{k!} \beta_k D^k$ (with $\beta_0 \neq 0$ ) one thus has

u_n(X) = \sum_{k=0}^{n} \binom{n}{k} \beta_k X^{n-k}.

(6)

One verifies that $u_n$ is a polynomial of degree $n$ (prop. 3); further

u_n(0) = \beta_n.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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