Inspired by this question and this answer, I decided to investigate the family of integrals
I(k)=∫e0Lik(lnx)dx,
where Lik(z) represents the polylogarithm of order k and argument z. I(1) evaluates to eγ, 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. ∫e0Li2(lnx)dx?=3F3(1,1,1;2,2,2;1)+π2(2e−5)12+γ22−γEi(1)=0.578255559804073275225659054377625577...
Brevan Ellefsen has computed that my conjecture is accurate to at least 150 digits. Brevan also gave the alternate form
π2e6+γG2,01,2(−1|10,0)+G3,02,3(−1|1,10,0,0).
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.
∞∑k=1I(k)=e∞∑k=2I(k)=e(1−γ)
PROGRESS UPDATE: Using this equation, I've turned 3F3(1,1,1;2,2,2;1) into
limc→1(Ei(1)−γc−1+1−e(c−1)2+(−1)−cΓ(c−1)c−1+(−1)1−cΓ(c,−1)(c−1)2),
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 3F3(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∈C,
G3,02,3(z|1,10,0,0)=γlnz+12ln2(z)−z3F3(1,1,1;2,2,2;−z)+γ22+π212.
My proof for this can be found here. Now I return to the limit (4).
Consider the following: 1c−1=c−1(c−1)2, (c−1)Γ(c−1)=Γ(c), and (−1)1−c=−(−1)−c. Based on these algebraic identities, the limit can be written as
limc→1(c−1)(Ei(1)−γ)+1−e+(−1)−c(Γ(c)−Γ(c,−1))(c−1)2.
In this form, the limit is 00. Using l'Hospital twice, we obtain
limc→1(−1)−c(−G4,03,4(−1|1,1,10,0,0,c)+Γ(c)(ψ0(c)22−iπψ0(c)+ψ1(c)2−π22))
Using Lemma (4.1), we know that
G3,02,3(−1|1,10,0,0)=3F3(1,1,1;2,2,2;1)+γ22+iγπ−5π212,
G3,02,3(−1|1,10,0,0)−γ22−iγπ+5π212=3F3(1,1,1;2,2,2;1).
Thus,
limc→1(Ei(1)−γc−1+1−e(c−1)2+(−1)−cΓ(c−1)c−1+(−1)1−cΓ(c,−1)(c−1)2)=3F3(1,1,1;2,2,2;1).
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