Legendre functions of the second kind (references)

Question

I would like to know where can I read a $\textbf{deduction}$ for the legendre functions of the second kind. I know that the Legendre polynomials (or Legendre functions of the first kind) are $P_\ell(x)$. If one tries to obtain a second solution, then this one should be proportional to $P_\ell(x)\ln x$ (by Fuchs theorem), however with the Legendre functions of the second kind we get a term $\ln\dfrac{x+1}{x-1}$. Where does this come from?

Answer 1

Not sure what Fuchs theorem implies in this case, but the general solution to the Legendre differential equation $[(1-x^2)y']' + n(n+1)y = 0$ is $y=\alpha P_n(x) + \beta Q_n(x)$ where $P_n(x)$ are the terminating series expansion Legendre polynomials, and $Q_n(x)$ are the infinite non-terminating series solutions, see for example About the Legendre differential equation.

You can see in that post that

$$Q_0(x) = \frac{1}{2} ln(\frac{1+x}{1-x})$$ And $$Q_1(x) = \frac{x}{2} ln(\frac{1+x}{1-x}) - 1$$ And the rest of the $Q_n$ can then be determined by using the recurrence relation $$(n+1)Q_{n+1}(x) = (2n+1)xQ_n(x) - nQ_{n-1}(x)$$

So all the $Q_n(x)$ contain terms containing a factor $ln(\frac{1+x}{1-x})$

That's where the factor / term comes from.