Mike Spivey

@robjohn: Just popping in to say that I noticed your answers to my old question. I don't have time to look closely at them now, but I will later (hopefully this afternoon). Thanks for posting them.

@OldJohn Either that, or we're both seeing things. :)

@JayeshBadwaik Following Old John's comment, I logged out, and there it was. I logged back in, and it was gone.

@JayeshBadwaik I see it on several other SE sites, too (also in Chrome) - just not on math.SE. Weird.

@JayeshBadwaik I see it on Stack Overflow, even in Chrome.

@JayeshBadwaik O.K., I see "About" when I use Firefox. But not with Chrome or Internet Explorer.

@JayeshBadwaik Interesting. Mine just has "Review Chat Meta Faq" across the top. No "About."

@JayeshBadwaik Where, exactly? I'm not seeing it.

@JasonBourne Thanks. I did find it, and I think it's a really good page, but if new users are going to get anything out of it then it needs to be obvious on the main site.

@robjohn: I noticed that there's now an "Informed" badge that you get for reading the entire "About" page. However, I can't find the link to the "About" page on the main site. It seems like that should be prominently displayed. Do you know anything about that?

@Tim: Thanks for the affirmation. :)

@OldJohn And I'm glad to hear I'm not the only one. :)

@OldJohn Re: Erdos and "simple arguments". I once needed the details of a two-line argument in a paper of Erdos. It took me two pages to fill everything in.

Sounds good. Best wishes! :)

You're probably right.

Yes. :)

I'd have to work through the details to make sure, but I'm pretty sure that it is.

It doesn't look like he uses those integral representations, though.

Well, this would only be the umpteenth time I've worked on something only to discover someone else has already done it. :)

I think you're right. :(

You're right. I guess ours is different.

The transformation $y_j = e^{u_j}$ gives the integral representation in eq. (3). That removes the log but introduces exp.

That's the transformation I was thinking of, too. I don't see yet how it removes the log. Maybe it isn't really the same as ours.

I'll have to take a closer look. I guess it needs a variable transformation to remove the log?

@Marvis Which formula?

Not sure. There are the Crandall, Borwein, and Zagier references, too.

Yes, it does.

Same here. I imagine we just need to work with them more.

For your work being new, I mean.

That's encouraging. :)

And the integrals I'm seeing involve logs. So they're more like the integral $\int_0^1 \log (1 + x) \log x \log (1- x) dx$ that Chris's sister asked about initially.

I'm not seeing the term "Euler sum" in his paper yet.

Or we could search for other articles that build more directly off of his so that we could find out exactly what is known about integral representations for Euler sums.

Or maybe your work could extend what he does.

And maybe your work is just repeating his.

@Marvis I just meant that if de Doelder has the same derivations you do then that's our first solid reference for your derivations.

Yes, thanks!

Does de Doelder have your integral representations?

No, I don't have access to it. (I miss Princeton's journal collection sometimes! I'm sure Stanford's is really good.)

Let me check.

@Marvis I would think so.

Here's a statement from that Flajolet and Salvy article: "Identities of low weight can sometimes be proved by special integral representations and functional properties of polylogarithms (de Doelder 1991)." So I guess we need to look up that de Doelder reference.

This article also has some: algo.inria.fr/flajolet/Publications/FlSa98.pdf. I verified one of the alternating Euler sums in my question with it.

Some of those conjectures may have been proven by now, as that article is almost 20 years old.

I don't know much about polylogs. I guess we would have to find out whether anyone else has used them to evaluate Euler sums.

There's a bunch of conjectures on page 26.

I think this article is the one that I found that had the most explicit evaluations: davidhbailey.com/dhbpapers/eulsum-em.pdf. See page 19.

Were you ever able to find out how to evaluate those other integrals in your answer (the ones that probably involved polylogs)?

Don't apologize! I think this one is the most promising, actually. :)

By the way, I think this is the third time you and I have discussed the possibility of writing a paper based on something from math.SE. :)