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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).

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).

(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-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 $W_1:=\frac{3}{2}z\\ .$

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).

(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 $W_1:=\frac{3}{2}z\\ .$

(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-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 $W_1:=\frac{3}{2}z\\ .$