Orthogonality relations for Legendre functions second kind Does someone know the orthogonality relations for the associated Legendre functions of the second kind $Q_{n}^{m}(z)$? Are they the same as the orthogonality relations $\int{P_{n}^{m}(z) P_{j}^{k}(z) dz}=\frac{2(n+m)!}{(2n+1)(n-m)!} \delta_{jn} \delta_{km}$ for the associated Legendre functions of the first kind?
 A: Given the imaginary unit modulus $i$, and integers $q$, and $p$, the orthogonality relationship for the associated Legendre functions of imaginary order for $x\in\mathbb{R}$ is
$$
 \int_{-1}^{1} Q^{i q}_{l}(x) Q^{i p}_{l}(x) \frac{dx}{1-x^{2}} = I^{q,p}_{3}
$$
where
$$
 I^{q,p}_{3} = 
\frac{\pi^{2}}
{2q \sinh (\pi q)}
\left(
\left( \cosh^{2} (\pi q) + 1\right) \delta^{q+p} +
     2 \cosh     (\pi q) 
\frac{\Gamma\left( 1 + l + i q\right)}
{\left( 1 + l + i q\right)}  \delta^{q-p}
\right)
$$
Reference: Orthogonality relations for the associated Legendre functions of
imaginary order, eqns 1.10, 3.25

The orthogonality relationships for the associated Legendre functions of the second kind with imaginary argument $Q^{m}_{n}(i x)$ with integer indices
$$
 \int_{\infty}^{\infty} Q^{k}_{m}(i x) Q^{k}_{n}(i x) dx =
\begin{cases}
  (2m)! \pi \left( \prod^{k}_{i=m+2} (m+1)(m-i+1)\right) \delta^{m}_{n} & k > m + 1 //
  (2m)! \pi  \delta^{m}_{n} & k = m + 1 //
\end{cases}
$$
with $k\in\mathbb{Z}_{+}$, $l\in\mathbb{N}$,$k>m$, and $x\in\mathbb{R}$.

ArXive reference: Orthogonality of the Ferrers’ Associated Legendre
Functions of the Second Kind with Imaginary
Argument
