Laplace's summation formula - MathOverflow

Laplace's summation formula

2010-10-24T03:22:58Z

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.

Answer by Gerry Myerson for Laplace's summation formula

2010-10-24T05:41:45Z

Did you try the Online Encyclopedia of Integer Sequences? http://www.research.att.com/~njas/sequences/A006232

Perhaps some of the references there will get you where you want to go.

Answer by SandeepJ for Laplace's summation formula

2010-10-25T14:04:23Z

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

Answer by Anixx for Laplace's summation formula

2010-10-28T12:55:52Z

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.

Answer by Tyler Clark for Laplace's summation formula

2010-11-20T16:06:58Z

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.