Example

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

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

If $U$ is the operator which to each polynomial $f(X)$ associates $f(X + \lambda)$ (where $\lambda \in K$ ), one has $U_0(X^k) = \lambda^k$ , and so $U = \sum_{k=0}^{\infty} \frac{1}{k!} (\lambda D)^k$ . By analogy with the series expansion of $e^{\lambda}$ (III, p. 105) we write $e^S$ or $\exp(S)$ for the formal series $\sum_{n=0}^{\infty} \frac{1}{n!} S^n$ in the ring $K[[S]]$ ; so one can write $U = e^{\lambda D}$ . Replacing the field $K$ by the field of rational fractions $K(Y)$ in this argument one sees similarly that the translation operator $T_Y$ can be written $e^{YD}$ .

Furthermore, in the ring $K[[S, T]]$ of formal series in two indeterminates over $K$ one has

(\exp S)(\exp T) = \sum_{p,q} \frac{S^p}{p!} \frac{T^q}{q!}

= \sum_{n=0}^{\infty} \frac{1}{n!} \left( \binom{n}{0} S^n + \binom{n}{1} S^{n-1} T + \cdots + \binom{n}{n} T^n \right)

= \exp(S + T)

and in particular

(\exp S)(\exp(-S)) = 1

which justifies the notation we have introduced.

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

Cited by

Nothing.

Cites

III, p. 105