An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as
$$\sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1) - f^{(2j - 1)}(0)],$$ where $B_{2j}$ is the $(2j)$th Bernoulli number. Here $n$ and $m$ are natural numbers and $f$ is a smooth enough function. A common case is when $m$ is somewhat large and $n$ is much greater than $m$.

An alternative summation formula is $$(*)\qquad\sum_{k=0}^{n-1}f(k) \approx\sum_{s=1-m}^{m-1}\tau_{m,1+|s|}\,\int_{s/2-1/2}^{n-1+1/2-s/2}f(x)\,dx,$$ where the $\tau_{m,j}$'s are certain coefficients depending only on $m$ and $j$, such that $\sum_{s=1-m}^{m-1}\tau_{m,1+|s|}=1$. More specifically, $$\tau_{m,r}:=2(-1)^{r-1}\,\sum_{\beta=0}^{\lfloor m/2-r/2\rfloor} \frac1{r+2\beta}\,\binom{2m}{m+r+2\beta}\Big/ \binom{2m}{m}.$$

Thus, the alternative formula approximates sums only by integrals of $f$, rather than by a mix of an integral of $f$ and values of $f$ and its derivatives up to a given order. Also, in contrast with the Euler--Maclaurin formula, the alternative approximating expression does not involve Bernoulli numbers. Both approximations, the Euler--Maclaurin and the alternative one, are exact when $f$ is any polynomial of degree at most $2m-1$.

I derived formula $(*)$ while working on a limit theorem in probability theory, where the Euler--Maclaurin formula did not seem to work well. Details and further discussion can be found [here].

Have you seen the alternative formula $(*)$ or a similar one elsewhere?

• books.google.com/books?id=laweX9bzDM8C – Steve Huntsman Nov 12 '15 at 2:19
• Thank you Steve for your comment. However, I am not sure what this reference means in relation to my question. In that book (by Beck and Robbins), I see a multidimensional version of the Euler--MacLaurin summation formula (which involves Bernoulli numbers, just as in the one-dimensional case). However, I don't see there in the book the alternative summation formula stated in my question (which does not involve Bernoulli numbers -- or derivatives -- in the approximating expression). Can you be more specific? – Iosif Pinelis Nov 12 '15 at 3:16
• I have added an explicit expression for the coefficients $\tau_{m,r}$ and comments about the origin of the alternative summation formula and (non)involvement of Bernoulli numbers. – Iosif Pinelis Nov 12 '15 at 4:55
• Sometimes functions are not so well behaved outside the interval of summation. Is there a version where the limits on the integrals step inside the interval of summation instead of outside? – Brendan McKay Nov 12 '15 at 5:19
• There are sevral variants of Euler--MacLaurin summation formula in the old book of J. F Steffensen "Interpolation" books.google.ru/books/about/… – Alexey Ustinov Nov 12 '15 at 6:09