Let $$I_n = \int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$$
In a recently published article, $I_n$ are evaluated for $n\leq 6$:
$$\begin{aligned}I_1 &= \frac{\log ^2(2)}{2}-\frac{\pi ^2}{12} \\ I_2 &= 2 \zeta (3) \log (2)-\frac{\pi ^4}{360}+\frac{\log ^4(2)}{4}-\frac{1}{6} \pi ^2 \log ^2(2) \\
I_3 &= \small 6 \zeta (3)^2+6 \zeta (3) \log ^3(2)-2 \pi ^2 \zeta (3) \log (2)+24 \zeta (5) \log (2)-\frac{23 \pi ^6}{2520}+\frac{\log ^6(2)}{6}-\frac{1}{4} \pi ^2 \log ^4(2)-\frac{1}{12} \pi ^4 \log ^2(2) \\
I_4 &= \small{-12 \pi ^2 \zeta (3)^2+288 \zeta (3) \zeta (5)+12 \zeta (3) \log ^5(2)-12 \pi ^2 \zeta (3) \log ^3(2)+168 \zeta (5) \log ^3(2)+108 \zeta (3)^2 \log ^2(2)-2 \pi ^4 \zeta (3) \log (2)-48 \pi ^2 \zeta (5) \log (2)+720 \zeta (7) \log (2)-\frac{499 \pi ^8}{25200}+\frac{\log ^8(2)}{8}-\frac{1}{3} \pi ^2 \log ^6(2)-\frac{19}{60} \pi ^4 \log ^4(2)-\frac{1}{6} \pi ^6 \log ^2(2)}
\end{aligned}$$
Based on these evidences, the author (me) made the conjecture that
For positive integer $n$, $I_n$ is in the algebra over $\mathbb{Q}$ generated by $\log(2)$ and $\{\zeta(m) | m\in \mathbb{Z}, m\geq 3\}$.
The closed-form of $I_5, I_6$ also satisfy this conjecture. $I_5$ is:
-20\pi^4\zeta(3)^2+7200\zeta(5)^2-960\pi^2\zeta(3)\zeta(5)+14400\zeta(3)\zeta(7)+20\zeta(3)\log^7(2)-40\pi^2\zeta(3)\log^5(2)+600\zeta(5)\log^5(2)+600\zeta(3)^2\log^4(2)-\frac{76}{3}\pi^4\zeta(3)\log^3(2)-560\pi^2\zeta(5)\log^3(2)+8640\zeta(7)\log^3(2)-360\pi^2\zeta(3)^2\log^2(2)+10080\zeta(3)\zeta(5)\log^2(2)+1440\zeta(3)^3\log(2)-\frac{20}{3}\pi^6\zeta(3)\log(2)-112\pi^4\zeta(5)\log(2)-2400\pi^2\zeta(7)\log(2)+40320\zeta(9)\log(2)-\frac{149\pi^{10}}{1320}+\frac{\log^{10}(2)}{10}-\frac{5}{12}\pi^2\log^8(2)-\frac{7}{9}\pi^4\log^6(2)-\frac{19}{18}\pi^6\log^4(2)-\frac{47}{60}\pi^8\log^2(2)
$I_6$ is:
10800\zeta(3)^4-100\pi^6\zeta(3)^2-36000\pi^2\zeta(5)^2-3360\pi^4\zeta(3)\zeta(5)-72000\pi^2\zeta(3)\zeta(7)+1123200\zeta(5)\zeta(7)+1209600\zeta(3)\zeta(9)+30\zeta(3)\log^9(2)-100\pi^2\zeta(3)\log^7(2)+1560\zeta(5)\log^7(2)+2100\zeta(3)^2\log^6(2)-140\pi^4\zeta(3)\log^5(2)-3000\pi^2\zeta(5)\log^5(2)+47520\zeta(7)\log^5(2)-3000\pi^2\zeta(3)^2\log^4(2)+90000\zeta(3)\zeta(5)\log^4(2)+24000\zeta(3)^3\log^3(2)-\frac{380}{3}\pi^6\zeta(3)\log^3(2)-2040\pi^4\zeta(5)\log^3(2)-43200\pi^2\zeta(7)\log^3(2)+739200\zeta(9)\log^3(2)-1140\pi^4\zeta(3)^2\log^2(2)+388800\zeta(5)^2\log^2(2)-50400\pi^2\zeta(3)\zeta(5)\log^2(2)+777600\zeta(3)\zeta(7)\log^2(2)-7200\pi^2\zeta(3)^3\log(2)-47\pi^8\zeta(3)\log(2)-560\pi^6\zeta(5)\log(2)+302400\zeta(3)^2\zeta(5)\log(2)-8880\pi^4\zeta(7)\log(2)-201600\pi^2\zeta(9)\log(2)+3628800\zeta(11)\log(2)-\frac{4714153\pi^{12}}{5045040}+\frac{\log^{12}(2)}{12}-\frac{1}{2}\pi^2\log^{10}(2)-\frac{37}{24}\pi^4\log^8(2)-\frac{253}{63}\pi^6\log^6(2)-\frac{527}{72}\pi^8\log^4(2)-\frac{223}{36}\pi^{10}\log^2(2)
Question: How to prove the conjecture for general $n$?
Any suggestion is appreciated.
Some remarks:
-
Even $I_3,I_4,I_5,I_6$ are extremely challenging, someone
brave enough might want to embark on finding them independently. -
$I_n$ is not related to beta function in an obvious way, so the
well-known differentiation trick does not work here. -
For any $I_n$, the algorithm outlined in the article should
produce closed-form of $I_n$ in a finite amount of time if the
conjecture is true. However, the algorithm is a bit mechanical, so
benefits little toward a proof for general $n$. - Perhaps I am missing something, this conjecture is elementary to
state, so it might have an easy proof and I was being negligent.
Best Answer
Denote $f(k,j)=\int_0^{\frac{1}{2}} \frac{\log ^j(1-y) \log ^k(y)}{1-y} \, dy$. Then for $j, k>1$ (RHS denotes Beta derivatives)
Which is direct by IBP, separation, Beta derivatives and reflection $y\to 1-y$: $$\small jf(k,j-1)= -(-\log (2))^{j+k}+ k \int_0^{\frac{1}{2}} \frac{\log ^j(1-y) \log ^{k-1}(y)}{y} \, dy$$ $$\small =-(-\log (2))^{j+k}+ k \left(\int_0^{1}-\int_{\frac{1}{2}}^1 \right) \frac{\log ^j(1-y) \log ^{k-1}(y)}{y} \, dy$$ $$\small =-(-\log(2))^{j+k}+ k \left( \partial_a^{k-1} \partial_b^j B\right) (0,1)-kf(j,k-1)$$ Thus taking $\frac{\binom{n-1}{j-1} \binom{n}{k}}{\binom{n}{j} \binom{n-1}{k-1}}=\frac{j}{k}$ into account yields the important $\color{blue}{formula}$
Now let $y\to\frac{1-x}{2}$ $$I_n=\int_0^{\frac{1}{2}} \frac{\log ^n(2 y) \log ^{n-1}(2 (1-y))}{1-y} \, dy$$ Apply Binomial thm twice, extract $k=0$ $$I_n=\sum _{k=1}^n \sum _{j=0}^{n-1} \binom{n}{k} \binom{n-1}{j} f(k,j) \log ^{2n-j-k-1}(2)+\int_0^{\frac{1}{2}} \frac{\log ^n(2) \log ^{n-1}(2 (1-y))}{1-y} \, dy$$ Take Cauchy product $$I_n=\sum _{m=1}^{2n-1} \sum _{k+j=m}\binom{n}{k} \binom{n-1}{j} f(k,j) \log ^{2n-m-1}(2)+\frac{\log ^{2 n}(2)}{n}$$ Take care of range of $j,k$ $$\scriptsize I_n=\sum _{m=1}^n \sum _{k=1}^m \binom{n}{k} \binom{n-1}{m-k} f(k,m-k) \log ^{2n-m-1}(2)+ \sum _{m=n+1}^{2 n-1} \sum _{k=m-n+1}^n \binom{n}{k} \binom{n-1}{m-k} f(k,m-k) \log ^{2n-m-1}(2)+\frac{\log ^{2 n}(2)}{n}$$ Let $k\to m+1-k$, take average $$\scriptsize I_n=\frac{1}{2} \sum _{m=1}^n \sum _{k=1}^m \left(\binom{n}{k} \binom{n-1}{m-k} f(k,m-k)+\binom{n}{m+1-k} \binom{n-1}{k-1} f(m+1-k,k-1)\right) \log ^{2n-m-1}(2)+\frac{1}{2} \sum _{m=n+1}^{2 n-1} \sum _{k=m-n+1}^n \left(\binom{n}{k} \binom{n-1}{m-k} f(k,m-k)+\binom{n}{m+1-k} \binom{n-1}{k-1} f(m+1-k,k-1)\right) \log ^{2n-m-1}(2)+\frac{\log ^{2 n}(2)}{n}$$ Use the $\color{blue}{formula}$ to simplify $$\scriptsize I_n=\frac{1}{2} \sum _{m=1}^n \sum _{k=1}^m \frac{\binom{n}{k}\binom{n-1}{m-k} \log ^{-m+2 n-1}(2) }{-k+m+1} U(k,m+1-k)+\frac{1}{2} \sum _{m=n+1}^{2 n-1} \sum _{k=m-n+1}^n \frac{\binom{n}{k}\binom{n-1}{m-k} \log ^{-m+2 n-1}(2) }{-k+m+1}U(k,m+1-k)+\frac{\log ^{2 n}(2)}{n}$$ Expand $U(k,m+1-k)$
This is the final expression of $I_n$. According to Lemma $2.3$ in OP's article, all Beta derivatives in this expression lie in the algebra $\mathbb{Q}(\pi^2, \zeta(3), \zeta(5), \zeta(7), \cdots)$, whence after adding up $\log(2)$ terms, $I_n$ lies in the extended $\mathbb{Q}(\log(2), \pi^2, \zeta(3), \zeta(5), \zeta(7), \cdots)$. QED.