I think, in order to be publishable, that the integral representations would need to lead to new evaluations of some Euler sums, though. And that's the part I'm not sure about.
Oh those are interesting set of conjectures! A quick glance at the conjectures : there is no one conjecture on $\sigma_h$ which is what the integral representation gives us
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.
Yes. Stanford has some good access to journal articles, one thing that I might miss after leaving this place. And yes I think it should be possible to evaluate $s_a$ if we have access to $\sigma_b$'s
Could you clarify what you mean by "Does de Doelder have your integral representations?"?
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.
I'm not seeing the term "Euler sum" in his paper yet.
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.
Somehow these polylogs, I am unable to get my head around it. There are a lot of interesting properties, but somehow I do not seem to have a good feel/understanding about these stuff.
It is not apparently clear to me as well. I too will have to take a closer look but looks like setting $y_j = 1/x_j$ and performing some cleaning up of the formula we have
The number of integral in this is only $k$, whereas ours has $s_1 + s_2 + \cdots s_k$ integrals. Mhenni's formula for instance collapses these integrals into one for $A(p,1)$
