How to directly prove an identity on infinite series

Question

$$ \begin{array}{c} \sum\limits_{l=0}^{\infty }C_{m+l}^{l}\frac{z^{x+l}-2^{-x-l}}{x+l}% =\sum\limits_{l=0,l\neq m}^{\infty }C_{l-x}^{l}\frac{2^{m-l}-\left( 1-z\right) ^{l-m}}{l-m}-C_{m-x}^{m}\ln (2-2z) \\ |z|,|1-z|<1,x\neq 0,-1,-2,\cdots ,m=0,1,2,\cdots, \end{array} $$ where $C_{l-y}^{l}=\frac{(1-y)_{l}}{l!},$ $\left( x\right) _{n}$ is a Pochhammer symbo, i.e. $\left( x\right) _{n}=x(x+1)\cdots (x+n-1)$.