# Z-transform and binomial series

I am reading a paper on frequency warping and I need to do a little manipulation of the Z-transform. Can somebody help me on how can I go about deriving equations $(3)$ and $(4)$ from equations $(1)$ and $(2)$.

$$\tag{1} \bar{A}(z) = \sum_{k=0}^{p}\bar{a}_k\bar{z}^{-k}(z), \ \ \ \ \ \ \bar{a}_0=1$$

$$\tag{2} \bar{z}^{-1} = \dfrac{(1-a^2)z^{-1}}{1-az^{-1}}-a$$

putting $(2)$ in $(1)$ and using binomial formulas:

$$\tag{3} \bar{A}(z) = \sum_{k=0}^{p}b_k\tilde{z}^{-k}(1-az^{-1})^{-k}$$

$$\tag{4} b_k=\sum_{n=k}^{p}C_{kn}\bar{a}_n, \ \ \ \ \ \ C_{kn}= {n \choose k}(1-a^2)^k(-a)^{n-k}$$