Closed form for $\int_0^e\mathrm{Li}_2(\ln{x})\,dx$?

Question

Inspired by this question and this answer, I decided to investigate the family of integrals $$I(k)=\int_0^e\mathrm{Li}_k(\ln{x})\,dx,\tag{1}$$ where $\mathrm{Li}_k(z)$ represents the polylogarithm of order $k$ and argument $z$. $I(1)$ evaluates to $e\gamma$, but $I(2)$ has resisted my efforts (which can be seen here).

Neither ISC nor WolframAlpha could provide a closed form for its numerical value--however, I've conjectured a possible analytic form. $$\eqalign{\int_0^e\mathrm{Li}_2(\ln{x})\,dx&\stackrel?=\,_3F_3(1,1,1;2,2,2;1)+\frac{\pi^2(2e-5)}{12}+\frac{\gamma^2}{2}-\gamma\,\mathrm{Ei}(1)\\&=0.578255559804073275225659054377625577...\tag{2}}$$

Brevan Ellefsen has computed that my conjecture is accurate to at least 150 digits. Brevan also gave the alternate form $$\frac{\pi^2e}{6}+\gamma G_{1,2}^{2,0}\left(-1\left|\begin{array}{c}1\\0,0\\\end{array}\right.\right)+G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right).\tag{2.1}$$

Is there a closed form for $I(2)$ that doesn't involve Meijer G or hypergeometrics? The simplicity of the following two equations seems to suggest that there might be. $(3.1)$ follows directly from $(3)$, which I've proven here. $$\begin{align} \sum_{k=1}^\infty I(k)&=e\tag{3}\\\sum_{k=2}^\infty I(k)&=e(1-\gamma)\tag{3.1}\end{align}$$

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PROGRESS UPDATE: Using this equation, I've turned $_3F_3(1,1,1;2,2,2;1)$ into $$\lim_{c\to 1}\left(\frac{\mathrm{Ei}(1)-\gamma}{c-1}+\frac{1-e}{(c-1)^2}+\frac{(-1)^{-c}\,\Gamma(c-1)}{c-1}+\frac{(-1)^{1-c}\,\Gamma(c,-1)}{(c-1)^2}\right),\tag{4}\label{4}$$ but I don't know how to proceed from there. EDIT: This limit leads nowhere. See below.

SECOND PROGRESS UPDATE: After some studying of the properties of the Meijer G function, I've finally cracked the limit; however, the result is an underwhelming $_3F_3(1,1,1;2,2,2;1)$. Before I evaluate the limit, I'd first like to state the following intermediate result:

Lemma $(4.1)$: For $z\in\mathbb{C}$, $$G_{2,3}^{3,0}\left(z\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)=\gamma\ln{z}+\frac12\ln^2(z)-z\,_3F_3(1,1,1;2,2,2;-z)+\frac{\gamma^2}2+\frac{\pi^2}{12}.\tag{4.1}\label{4.1}$$

My proof for this can be found here. Now I return to the limit $\eqref{4}$. Consider the following: $\frac{1}{c-1}=\frac{c-1}{(c-1)^2}$, $(c-1)\Gamma(c-1)=\Gamma(c)$, and $(-1)^{1-c}=-(-1)^{-c}$. Based on these algebraic identities, the limit can be written as $$\lim_{c\to 1}\frac{(c-1)(\mathrm{Ei}(1)-\gamma)+1-e+(-1)^{-c}(\Gamma(c)-\Gamma(c,-1))}{(c-1)^2}.\tag{4.2}$$ In this form, the limit is $\frac{0}0$. Using l'Hospital twice, we obtain $$\lim_{c\to 1}{(-1)^{-c}\left(-G_{3,4}^{4,0}\left(-1\left|\begin{array}{c}1,1,1\\0,0,0,c\\\end{array}\right.\right)+\Gamma{(c)}\left(\frac{\psi_0{(c)}^2}2-i\pi\psi_0(c)+\frac{\psi_1(c)}2-\frac{\pi^2}2\right)\right)}$$ $$\begin{align}&=G_{3,4}^{4,0}\left(-1\left|\begin{array}{c}1,1,1\\0,0,0,1\\\end{array}\right.\right)-\frac{\psi_0(1)^2}2+i\pi\psi_0(1)-\frac{\psi_1(1)}2+\frac{\pi^2}2\\&=G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array} \right.\right)-\frac{\gamma^2}{2}-i\gamma\pi+\frac{5\pi^2}{12}.\tag{4.2a}\end{align}$$ Using Lemma $\eqref{4.1}$, we know that $$G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)={}_3F_3(1,1,1;2,2,2;1)+\frac{\gamma^2}2+i\gamma\pi-\frac{5\pi^2}{12},\tag{4.3}$$ which can be rewritten as $$G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)-\frac{\gamma^2}{2}-i\gamma\pi+\frac{5\pi^2}{12}={}_3F_3(1,1,1;2,2,2;1).\tag{4.3a}$$

Thus, $$\eqalign{&\lim_{c\to 1}\left(\frac{\mathrm{Ei}(1)-\gamma}{c-1}+\frac{1-e}{(c-1)^2}+\frac{(-1)^{-c}\,\Gamma(c-1)}{c-1}+\frac{(-1)^{1-c}\,\Gamma(c,-1)}{(c-1)^2}\right)\\=&{}_3F_3(1,1,1;2,2,2;1).\tag{4.4}}$$

Answer 1

Here I would like to sketch an alternative integral representation for $$ I(2)=\int_0^e\operatorname{Li}_2\left(\ln{x}\right)\,dx. $$ I hope that this argument shed some light on, why we cannot expect a much better representation as the hypergeometric series representation, which was given in the question.

After a change of variable $x \mapsto \ln(x)$, using an integral representation for $\operatorname{Li}_2$, if we interchange of the order of integration, we arrive at \begin{align} I(2) = \int_0^e\operatorname{Li}_2\left(\ln{x}\right)\,dx &= \int_{-\infty}^1e^x\operatorname{Li}_2\left(x\right)\,dx \\ &= - \int_{-\infty}^1 e^x \int_0^1\frac{\ln\left(1-xt\right)}{t}\,dt\,dx \\ &= - \int_0^1 \frac{1}{t} \int_{-\infty}^1 e^x\ln\left(1-xt\right)\,dx\,dt. \end{align}

By using the antiderivative of the function $e^x\ln\left(1-xt\right)$ with respect to $x$, we have \begin{align} I(2) &= - \int_0^1 \left(e\ln(1-t) + e^{1/t}E_1\left(\frac{1}{t}-1\right)\right)\,\frac{dt}{t} \\ &= - e\int_0^1 \frac{\ln(1-t)}{t} \,dt - \int_0^1 e^{1/t}E_1\left(\frac{1}{t}-1\right)\,\frac{dt}{t} \\ &= \frac{e\pi^2}{6} - \int_0^1 e^{1/t}E_1\left(\frac{1}{t}-1\right)\,\frac{dt}{t}, \end{align} where $E_1(x)$ is the $E_n$ function of order $n=1$. Using a change of variable $t\mapsto1/x$, we have $$ I(2) = \frac{e\pi^2}{6} - \int_1^\infty \frac{e^x E_1(x-1)}{x}\,dx. $$ We stop here. Although it would be nice to evaluate this integral, but as I know, this kind of integrals are not in the literature. Here I want to mention an earlier identity in a question of mine, where a solution of a related integral problem was also given in terms of ${_k}F_k\left(1,\dots,1;2,\dots,2;1\right)$. Of course any ideas for further evaluation are welcome. By using an integral representation for the $E_1$ function, we arrive at \begin{align} I(2) &= \frac{e\pi^2}{6} - \int_1^\infty \frac{e^x}{x} \int_1^\infty \frac{e^{-(x-1)t}}{t} \,dt \,dx \\ &= \frac{e\pi^2}{6} - \int_1^\infty \int_1^\infty \frac{e^{x+(1-x)t}}{xt} \,dt \,dx. \end{align}

Answer 2

Too long for a comment.

If we evaluate the decay rate of your sum we find that $$\sum_{k=1}^\infty\frac{1}{k^2 k!}<\frac{1}{\sqrt{2\pi}}\sum_{k=1}^\infty\frac{e^k}{k^{k+5/2}}<\frac{1}{\sqrt{2\pi}}\sum_{k=1}^\infty\left(\frac{e}{k}\right)^k$$ This final series goes to zero smaller than the function $x^{-n}$ for any positive $n$, which helps us estimate the decay rate a little. We can also note that a number is likely irrational if $a_{k+1} \ll a_k^2$. Here we have that $$a_k^2\sim\left(\frac{1}{\sqrt{2\pi}\;k^{k+ 5/2} e^{-k}}\right)^2=\frac{e^{2 k} k^{-2 k - 5}}{2 \pi}$$ $$a_{k+1} \sim \frac{1}{\sqrt{2\pi}\;(k+1)^{n+7/2} e^{-k-1}} = \frac{e^{k + 1} (k + 1)^{-k - 7/2}}{\sqrt{2 \pi}}$$ This tells us that $$\frac{a_k^2}{a_{k+1}} \sim \frac{e^{k - 1} k^{-2 k - 5} (k + 1)^{k + 7/2}}{\sqrt{2 \pi}}$$ It's interesting to note how fast this goes to zero; in fact, if we denote the above by $f(k)$ and let $n$ be any real number we find that $$\lim_{k\to\infty}\frac{f(k)}{e^{nk}}=0$$
Thus, we see that this decay is very rapid, and conclude that $a_{k+1} \ll a_k^2$. I'm pretty sure this implies a proof of irrationality could be made (the fact that it is a sum of rational, decreasing terms alone encourages this), and possibly even a proof that the series is transcendental. Not too keen with constructing sequences to prove even irrationality though. Just let these serve as heuristics for now!