Euler-MacLaurin総和公式の「導出」

Wikipediaには色んなことが書いてあるのう...

Since the summation operator Σ is the inverse operator to the difference operator Δ we get
$\Sigma = \Delta^{-1} = \frac1{e^D - I}$ .
Now we know that the exponential generating function of the Bernoulli numbers is given by
$\frac{x}{e^x-1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}$ ,
hence formally
$\Sigma = \frac1D \sum_{n=0}^\infty B_n \frac{D^n}{n!} = \frac1D -\frac12 I + \frac{1}{12} D + \cdots = \int - \frac12 I + \frac{1}{12} D + \cdots$ ,
where ∫ denotes the integral operator. This purely formal derivation indicates the existence of the formula. The idea is due to Legendre.

Bernoulli多項式の母関数は忘れそうだけど，そこだけどうにかすればEuler-MacLaurin総和公式を暗記できるな．
もうちょっとしっかりした導出はこちら：

http://www.math.ntnu.no/~bryn/TMA4215/current/notater/EulerMacLaurin.pdf NOTES ON THE EULER-MCLAURIN SUMMATION FORMULA