I recently came across the following formula, which is apparently known as *Laplace's summation formula:*

$$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - \frac{1}{12} \left(\Delta f(b) - \Delta f(a)\right) $$ $$+ \frac{1}{24} \left( \Delta^2 f(b) - \Delta^2 f(a) \right) - \frac{19}{720} \left(\Delta^3 f(b) - \Delta^3 f(a) \right) + \cdots$$

(Of course, the right-hand side isn't guaranteed to converge.) The coefficient on the term with $\Delta^{k-1}$ is $\frac{c_k}{k!}$, where $c_k$ is apparently called either a *Cauchy number of the first kind* or a *Bernoulli number of the second kind*.

The formula looks to me like a finite calculus version of the Euler-Maclaurin summation formula.

I'm trying to find out more about Laplace's summation formula. However, the usual suspects (the arXiv, Wikipedia, MathWorld, Google) aren't turning up much. There was a little on MathSciNet, the most promising of which was a paper by Merlini, Sprugnoli, and Verri entitled "The Cauchy Numbers" (*Discrete Mathematics* 306(16): 1906-1920, 2006). The MathSciNet review says, "Application of the Laplace summation formula involving the harmonic numbers [is] also given." I've requested the paper through interlibrary loan, but it has not arrived yet.

While I'm interested in the formula in general, I'm particularly interested in these two questions.

What applications are there for the Laplace summation formula? (It seems like there ought to be a sufficient number of applications for it to deserve having Laplace's name attached to it. I suppose one could use it for asymptotic analysis, but I'm not sure what the advantage would be over Euler-Maclaurin.)

What is the error bound on the formula when it is truncated after $n$ terms?

I wasn't sure how to tag this; feel free to retag.