Formula for the derivative of finite power series in reversed order of terms.

Question

I wanted to solve the polar part in Schrödinger's wave equation for the H-atom by direct substitution of functions of form:- $$ \Theta_{lm}(\theta) = a_{lm} \sin^{|m|}\theta \sum_{r≥0}^{r≤(l-|m|)/2}(-1)^rb_r \cos^{l-|m|-2r}\theta $$ The $a$'s are normalisation constants, no problem there. However, the problem of determining the $b$'s ultimately drops down to finding the first and second derivatives of the polynomial in $z=\cos \theta$: $$ P(z)=\sum_{r≥0}^{r≤(l-|m|)/2} (-1)^rb_r z^{l-|m|-2r} $$ Which is a finite power series written in decreasing order of powers. I couldn't find a formula for so (well sometimes I get that dumb), but I think it does exist, maybe some reference book or website. I emphasize that what I'm doing is right the reverse of Frobenius-method. Thanks.