Associate Legendre polynomials of first and second kind; the integral relastionship

Question

The Legendre functions of first $P_n(x)$ and second $Q_n(x)$ kind are related by the definite integral $$ Q_n(x) = {1\over 2} \int_{1}^{-1}{P_n(x) \over u-x}\, du. $$ The associated Legendre functions ($P_n^m$ and $Q_n^m$) are related to ordinary Legendre functions ($P_n$ and $Q_n$) for $|x|>1$ by $$ P_n^m(x) = (x^2-1)^{m/2}{d^m \over dx^m}P_n(x),\qquad Q_n^m(x) = (x^2-1)^{m/2}{d^m \over dx^m}Q_n(x), $$ Now, is there a definite integral expression that relates the "associate Legendre functions" of first $P_n^m(x)$ and second $Q_n^m(x)$ kind?

Answer 1

A. Erdélyi et al., Higher Transcendental Functions Vol. I, section 3.6.2 formula (32) list the following Generalization of Neumann's formula (attributed to Gormley, 1934) $$Q_{\nu}^{\mu}(z) = \frac{1}{2}e^{i\mu\pi}(z^2-1)^{\mu/2}\int_{-1}^{1}(1-v^2)^{-\mu/2}\frac{P_{\nu}^{\mu}(v)}{z-v} dv$$

with the restrictions $\nu+\mu = 0,1,\dots,\;\Re \nu > -1,\; z \not \in [-1,1]$