for any n is positive integer LegendreP function can be expressed as $\displaystyle P_n(x)=\frac{1}{2^n n!}\frac{d^n}{dx^n}\left[(x^2-1)^n\right]$.
Let $\displaystyle q_n(x)=Q_n(x)-P_n(x)\log\left(\frac{1+x}{1-x}\right)$, then $q_0=0,q_1=-1,(n+1)q_{n+1}=(2n+1)xq_n-nq_{n-1}$, is it also possible to simplify $q_n$ by using Rodrigues' formula or other "elementary form"?
$\displaystyle q_n(x)=P_n(x)\int_{\infty}^x \frac{P_n(t)^2-1}{P_n(t)^2(t^2-1)}dt$
From Relationship between Legendre polynomials and Legendre functions of the second kind