AMZV
$$
\sum _{0< n,m}\frac {1}{n^{k+1}(n+m)}\frac {n!m!}{(n+m)!}
=\zeta (k+2)+2\zeta (\bar {k},\bar 2)
$$
$$
\begin {aligned}
&\sum _{0< n,m}\frac {1}{n^{k+1}(n+m)}\frac {n!m!}{(n+m)!}\\
&=\sum _{0< n,m}\frac {1}{n^{k}}\int _{0}^{1}y^{n+m-1}dy\int _{0}^{1}x^{n-1}(1-x)^{m}dx\\
&=\iint _{0< x,y< 1}\sum _{0< n}\frac {(xy)^{n}}{n^k}\sum _{0< m}(y(1-x))^{m}\frac {dx}x\frac {dy}y\\
&=\iint _{0< x,y< 1}\frac {\operatorname{Li} _{k}(xy)y(1-x)} {(1-y(1-x))xy}dxdy\\
&=\iint _{0< x< y< 1}\frac {\operatorname{Li} _{k}(x)(y-x)} {(1-y+x)xy}dxdy\\
&=\iint _{0< x< y< 1}\frac {\operatorname{Li} _{k}(x)}{x(1+x)}\left (\frac {y-x}{y}+\frac {y-x}{1+x-y}\right )dxdy\\
&=\iint _{0< x< y< 1}\frac {\operatorname{Li} _{k}(x)} {x(1+x)}\left (-\frac {x}y+\frac {1}{1+x-y}\right )dxdy\\
&=\int _{0}^{1}\frac {\operatorname{Li} _{k}(x)}{x(1+x)}\left (-x\ln
\frac {1}x+\ln \frac {1}{x}\right ) dx\\
&=\sum _{0< n,m}\int _{0}^{1}\frac {x^{n-1}(-x)^{m-1}}{n^{k}}\left (-x\ln \frac {1}x+\ln \frac {1}x\right )dx\\
&=\sum _{0< n,m}\frac {(-1)^{m}}{n^k}\left (\frac {1}{(n+m)^2}-\frac {1}{(n+m-1)^2}\right )\\
&=\zeta (\bar {k},\bar 2)+\zeta ^{\star}(\bar {k},\bar 2)\\
&=\zeta (k+2)+2\zeta (\bar {k},\bar 2)
\end {aligned}
$$
Level 4
$$
\begin {aligned}
z\int _{0}^{1}\frac {\ln ^n\frac {1}{1-x}\ln ^{m}\frac {1}x}{1-zx}dx&=\int _{0< u_1,\cdots ,u_n< x< v_1,\cdots ,v_m< 1}\left (\frac {du}{1-u}\right )^n\left (\frac {dv}v\right )^m\frac {zdx}{1-zx}\\
&=n!m!\int _{0< u_1<\cdots < u_n< x< v_1<\cdots < v_m< 1}\left (\frac {du}{1-u}\right )^n\frac {zdx}{1-zx}\left (\frac {dv}v\right )^m\\
&=n!m!\sum _{0< r_1<\cdots < r_n}\frac {1}{r_1\cdots r_n}\int _{0< x< v_1<\cdots < v_m< 1}\frac {zx^{r_n}dx}{1-zx}\left (\frac {dv}v\right )^m\\
&=n!m!\sum _{0< r_{1}<\cdots < r_{n}}\frac {1}{r_{1}\cdots r_{n}}\sum _{0< k}z^{k}\int _{0< x< v_1<\cdots < v_m< 1}x^{r_n+k-1}dx\left (\frac {dv}v\right )^m\\
&=n!m!\sum _{0< r_{1}<\cdots < r_{n}}\frac {1}{r_{1}\cdots r_{n}}\sum _{0< k}\frac {z^k}{\left (r_1+k\right )^{1+m}}\\
&=n!m!\sum _{0< r_{1}<\cdots < r_{n+1}}\frac {z^{r_{n+1}-r_n}}{r_{1}\cdots r_{n}r_{n+1}^{m+1}}
\end {aligned}
$$
↓合ってるか分からない
$$
\begin {aligned}
&
\Im \sum _{0\leq j\leq k\leq n}\sum _{0\leq r\leq m}\left (\pi i\right )^{n+m-k-r}\binom kj\binom nk\binom mr\left (-1\right )^{j+k+r}\int _{0}^{1}\ln ^{k-j}\frac {1}{1-x}\ln ^{r+j}\frac {1}x\frac {dx}{1+x^{2}}\\
\\
&\overset{?}=\pi \Im \left (\left (\frac {\ln 2}2+\frac {\pi i}4\right )^n\left (\frac {\pi i}2\right )^m\right )
\end {aligned}
$$