Hermite identity

An identity applied by Ch. Hermite (1873) to certain specially constructed polynomials in the proof that the number $e$ is transcendent. In a simplified form it is $$ e^x F(0) - F(x) = e^x \int_0^x e^{-t} f(t) dt $$ where $f(x)$ is a polynomial in $x$ and $$ F(x) = \sum_{k=0}^\infty f^{(k)}(x) \ . $$

Comments

For a proof of the transcendence of $e$ and use of Hermite's identity see, e.g., [a1], Thm. 6.4. There a simplified version of Hermite's original proof is given.

References

[a1]	I. Stewart, "Galois theory", Chapman & Hall (1979) Zbl 0269.12104