Proposition 4

Tag 08I0, permanent. Cite it as [Bourbaki, Tag 08I0].

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

The Appell polynomials attached to $U$ satisfy the relations

\frac{d u_n}{d X} = n \, u_{n-1}

(7)

u_n(X + Y) = \sum_{k=0}^{n} \binom{n}{k} u_{n-k}(X) Y^k

(8)

U(u_n(X)) = \frac{n!}{(n-p)!} X^{n-p}.

(9)

These formulae are in fact respectively equivalent to the following relations (bearing def. 2 in mind):

DV^{-1} = V^{-1}D \tag{10}

(\exp(YD_X))V_X^{-1} = V_X^{-1}\exp(YD_X) \tag{11}

UV^{-1} = D^p. \tag{12}

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

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