Laplace's summation formula 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.
 A: 
However, the usual suspects (the
  arXiv, Wikipedia, MathWorld, Google)
  aren't turning up much.

You forgot google books!
There are references to the Laplace summation formula in two books.  


*

*Page 248 in The rise and development of the
theory of series up to the early
1820s by Giovanni Ferraro
http://books.google.com/books?id=vLBJSmA9zgAC

*Page 192 in A history of numerical analysis from
    the 16th through the 19th century by
    Herman Goldstine
    http://books.google.com/books?id=20csAQAAIAAJ
A: You may be also interested in this formula for indefinite sum of $f(x)$:
$$\sum_x f(x)=-\sum_{k=1}^{\infty}\frac{\Delta^{k-1}f(x)}{k!}(-x)_k+C$$
where $(x)_k=\frac{\Gamma(x+1)}{\Gamma(x-k+1)} $ is a falling factorial.
A: Did you try the Online Encyclopedia of Integer Sequences? 
http://oeis.org/A006232 
Perhaps some of the references there will get you where you want to go. 
A: This is a bit late - I could be completely wrong, but I think the issue here is the domain being used.
Laplace's summation formula should be used on the set of integers and will be used for calculations in discrete calculus. I believe that the Euler-Maclaurin summation formula is typically used on the reals though.
I hope this helps.
