I am trying to compute the first moment using the Laplace transform of the following expression...
$$P^*(s,z)=\frac{z^{i+1} - z^h(1-z)(\mu-\lambda z)P_h^*(s)}{sz-(1-z)(\mu-\lambda s)}$$,
where,
$$P_h^*(s) =\alpha_2^{i-h}$$,
and
$$\alpha_2=\frac{\lambda +\mu +\sqrt{(\lambda +\mu +s)^2-4 \lambda \mu }+s}{2 \lambda }$$
I understand that I have to take the derivative, then set $s=0$. But the problem is that this expression has a "z" in it. What do I do with that? This appears to be some sort of joint transform or something. Can I just ignore the "z" (i.e., set it to 1)?
Any help would be appreciated...