Legendre Polynomial Of Second Kind

Question

How to calculate the normalisation factors of Legendre Polynomial of second kind? It is provided that ,the normalisation factors are chosen so that second kind Polynomials satisfies the recurrence relation of the first kind.

Answer 1

Assuming you mean for $-1 < x < 1$ and you are not considering complex arguments.

The function of the 2nd kind $Q_n(x)$ is not a polynomial.

It is not clear what normalisation factors you are referring to but the functions of the 2nd kind can be computed as follows.

It satisfies the same recurrence relation as the Legendre polynomial $P_n(x)$.

Initial values for the recursion are $$ Q_0(x)=\frac{1}{2}\ln\left( \frac{1+x}{1-x} \right) $$ and $$ Q_1(x)=\frac{x}{2}\ln\left( \frac{1+x}{1-x} \right)-1 $$ See for example here