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\$:
$$(n+1)W_n=(2n+1)zW_{n-1}-nW_{n-2}\qquad (n\ge2)$$$$with the initial contitions W_0=1 and W_1:=\frac{3}{2}z\ . edit. Note that one may start the above recurrence with W_{-1}:=0 and W_0:=1. Also note that, up to a shift, that recurrence is the same as the Legendre polynomials. This means that the polynomials R_n:=W_{n-1} are the other linear independent solution to the recurrence of the Legendre polynomials,$$(n+1)y_{n+1}=(2n+1)zy_{n}-ny_{n-1}\ ,$$that corresponds to the initial conditions R_0=0, R_1=1 (while P_0=1 and P_1=z). According to the notations of the general theory of orthogonal polynomials, these R_n should be named "Legendre polynomials of the second kind" (not to be confused with the Legendre functions of the second kind). 3 deleted 1 characters in body (to question 2). The generating function of the sequence W_n(z) (shifted) is$$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$$that translates into a three-terms three-term linear recurrence for the W_n\ :$$(n+1)W_n=(2n+1)zW_{n-1}-nW_{n-2}\qquad (n\ge2)$$with the initial contitions W_0=1 and W_1:=\frac{3}{2}z\ . 2 added 17 characters in body (to question 2). The generating function of the sequence W_n(z) (shifted) is$$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$$that translates into a three-terms linear recurrence for the W_n\ :$$(n+1)W_n=(2n+1)zW_{n-1}-nW_{n-2}\qquad (n\ge2)
with the initial contitions $W_0=1$ and $W_1:=\frac{3}{2}z\ .$