 Legendre function of the second kind

For nonnegative integer $n$, a second solution of the Legendre differential equation is obtained by reduction of order from $P_n$ as $P_n(x)\int dx/[(1-x^2)P_n(x{)}^2]$. Normalize it as $Q_0=\frac{1}{2}\log |(1+x)/(1-x)|$ and $Q_1=x{Q}_0-1$, with the usual Legendre polynomial recurrence relation for higher degrees. Inside $(-1,1)$, $Q_n\sim -\frac{1}{2}\log (1-x)$ near $1$ and $Q_n\sim (-1{)}^n\frac{1}{2}\log (1+x)$ near $-1$. These follow from $P_n(1)=1$, $P_n(-1)=(-1{)}^n$ in the integral. Poles in the integral at interior polynomial zeros cancel in the product, by reduction of order across a zero of the known solution. On $x>1$, the integral from infinity defines the decaying normalization $Q_n(x)\sim 2^n(n!{)}^2{x}^{-n-1}/(2n+1)!$. For $n\ge 1$ all bounded exterior solutions decay this way; for $n=0$ an additional constant solution remains bounded.