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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.

  1. 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.)

  2. 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.

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Is the following paper any useful: – Suvrit Oct 24 '10 at 15:44
@Suvrit: It looks very useful. In particular, it appears to have the remainder expression I was hoping for. Thanks! – Mike Spivey Oct 24 '10 at 19:37
The place to look is probably "Geschichte der Zeta-Funktion von Oresme bis Poisson" by Georg Schuppener. Unfortunately, it is very hard to find (besides being in German). If I recall it correctly, he carefully goes through the history of these kind of summation formulas. – Franz Lemmermeyer Oct 25 '10 at 14:26
@Franz: Thanks. I'll try to track it down. With my somewhat passable reading German (at least in mathematics) and help from the German in the office next to mine, I should be O.K. with the language issue. – Mike Spivey Oct 25 '10 at 17:22

Did you try the Online Encyclopedia of Integer Sequences?

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

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Thanks. I did look at the OEIS; I should have mentioned that in my original post. I have not tracked down those references yet, but I will. – Mike Spivey Oct 24 '10 at 19:41

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.

  1. Page 248 in The rise and development of the theory of series up to the early 1820s by Giovanni Ferraro

  2. Page 192 in A history of numerical analysis from the 16th through the 19th century by Herman Goldstine

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Thanks. I had found the Giovanni text, but there isn't much more there other than the formula itself. (Giovanni also has an error in the formula). I'll check out the Goldstine book. – Mike Spivey Oct 25 '10 at 15:14

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.

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The unique $n$ should be $k$. – JBL Oct 28 '10 at 13:14
Yes, corrected. – Anixx Oct 28 '10 at 13:18

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.

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