Discovering and Proving Infinite Pochhammer Sum Identities

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Table

$l_2:=\hspace{0.17em}\text{log}\hspace{0.17em}(2)$	$l_3:=\hspace{0.17em}\text{log}\hspace{0.17em}(3)$	${\zeta }_3:=S_3(\infty );$
${\zeta }_5:=S_5(\infty );$	${\zeta }_7:=S_7(\infty );$	${\zeta }_9:=S_9(\infty );$
${\zeta }_{11}:=S_{11}(\infty );$	$C:=\text{Catalan};$	$p_4:={\text{Li}}_4\left(\frac{1}{2}\right);$
$p_5:={\text{Li}}_5\left(\frac{1}{2}\right);$	$p_6:={\text{Li}}_6\left(\frac{1}{2}\right);$	$p_7:={\text{Li}}_7\left(\frac{1}{2}\right);$
$p_8:={\text{Li}}_8\left(\frac{1}{2}\right);$	$p_9:={\text{Li}}_9\left(\frac{1}{2}\right);$	$s_1:=S_{-5,-1}(\infty );$
$s_2:=S_{5,-1,-1}(\infty );$	$s_3:=S_{-5,1,1}(\infty );$	$s_4:=S_{5,3}(\infty );$
$s_5:=S_{-7,-1}(\infty );$	$s_6:=S_{-5,-1,-1,-1}(\infty );$	$s_7:=S_{-5,-1,1,1}(\infty );$
$h_1:=H_{(3,0),(0,0)}(1);$	$h_2:=H_{(3,0),(0,0),(1,0)}(1);$	$h_3:=H_{(3,0),(0,0),(0,0),(0,0)}(1);$
$h_4:=H_{(3,0),(0,0),(1,0),(1,0)}(1);$	$h_6:=H_{(5,1)}(1);$	$h_6:=H_{(5,3)}(1);$
$h_7:=H_{(5,1),(0,0)}(1);$	$h_8:=H_{(5,2),(0,0)}(1);$