The generating function of the sequence $W_n(z)$ (shifted) is
$$w(t,z):=(1-2zt+t^2)^{-\frac{1}{2}}\log\bigg(\frac{-z+t+(1-2zt+t^2)^\frac{1}{2}}{1-z}\bigg) =\sum_{n=1}^\infty\\  W_{n-1} (z)\\ t^n\\ .$$
It verifies a simple linear first order differential equation:
$$(1-2zt+t^2)w_t + (t-z)w=1$$
that translates into a three-term linear recurrence for the $W_n\\ $:

$$(n+1)W_n=(2n+1)zW_{n-1}-nW_{n-2}\qquad $$

with the initial contitions $W_0=1$ and $W_1:=\frac{3}{2}z\\ .$ 



**edit**. Note that one may start the above recurrence with $W_{-1}:=0$ and $W_0:=1$. Also note that, up to a shift, that recurrence is the same as  the Legendre polynomials. This means that the polynomials $R_n:=W_{n-1}$ are the other linear independent solution to the recurrence of the Legendre polynomials,
$$(n+1)y_{n+1}=(2n+1)zy_{n}-ny_{n-1}\\ ,$$
that corresponds to the initial conditions $R_0=0$, $R_1=1$ (while $P_0=1$ and $P_1=z$). According to the notations of the general theory of orthogonal polynomials, these $R_n$ should be named "*Legendre polynomials of the second kind*" (not to be confused with the Legendre *functions* of the second kind).