The moments of come from those of , and involves sums like for even $m$, and the moments of a uniform distribution between -1/2 and 1/2. Perhaps someone knows formulas for these.

The m’th derivative of (for ) can be found (I think!) by replacing of the by its derivatives , and those are made of Dirac delta functions at something like . The remaining part has to be evaluated at these points.

]]>Ah, I see! Thanks for that note, that makes sense.

]]>Convolutions are definitely the key to understanding this problem! I like your line of thinking. But there’s a nuance here. For the original Borwein integrals

we are interested in how far falls. But for the *cosine* Borwein integrals

we are interested in how far falls. We need to reach for to start falling, but we need to reach for to start falling. All this is explained by Greg Egan here:

• Patterns that eventually fail, *Azimuth*, September 18, 2018.

It has some value, especially if you’re using the math to build bridges or something like that. But in pure math it really just sets the stage for what I consider the *interesting* part, namely understanding things.