Proposition 5

Tag 08I1, permanent. Cite it as [Bourbaki, Tag 08I1].

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

For every polynomial $f \in K[X]$ and every composition operator of order $p$ one has

f^{(p)}(X + Y) = \sum_{k=0}^{\infty} \frac{1}{k!} U \left( f^{(k)}(X) \right) u_k(Y) \tag{13}

(generalized Taylor expansion).

Indeed, if one puts $U = D^p V = V D^p$ one has (VI, p. 270, formula (1))

V_X^{-1}(f(X + Y)) = V_Y^{-1}(f(X + Y)) = \sum_{k=0}^{\infty} \frac{1}{k!} f^{(k)}(X) u_k(Y) \tag{14}

the Taylor formula and by def. 2 of VI, p. 273; it suffices to apply the operator

U

to the first and last terms of formula (14) to obtain (13).

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

Cited by

Nothing.