I am trying to find the following recurrence relation for these polynomials concerning its derivative:
$$(1-x^2)\frac{dP_l^m}{dx}=-lxP_l^m+(l+m)P_{l-1}^m$$
employing the generating function:
$$T_m(x,s)=\frac{(2m)!(1-x^2)^{m/2}s^m}{2^mm!(1-2xs+x^2)^{m+1/2}}=\displaystyle\sum_{l=m}^\infty P_l^ms^l$$
and the next relation: $$(2l+1)xP_l^m=(l-m+1)P_{l+1}^m+(l+m)P_{l-1}^m$$
but the only expression I get is one concerning two derivatives (which also seems to be wrong):
$$(1-x^2)\frac{2m-1}{l+m}\frac{dP_{l+1}^m}{dx}+(1-x^2)x\frac{(1-2m)}{l+m}\frac{dP_l^m}{dx}=\\ =-mxP_{l+1}^m+(2m-x^2+1-\frac{(2l+1)(1-x^2)}{l+m})P_l^m-mxP_{l-1}^m$$
Could anyone help me?