I Associated Legendre polynomials: complex vs real argument

[avikarto]

I am having trouble understanding the relationship between complex- and real-argument associated Legendre polynomials. According to Abramowitz & Stegun, EQ 8.6.6,
$$P^\mu_\nu(z)=(z^2-1)^{\mu/2}\cdot\frac{d^\mu P_\nu(z)}{dz^\mu}$$
$$P^\mu_\nu(x)=(-1)^\mu(1-x^2)^{\mu/2}\cdot\frac{d^\mu P_\nu(x)}{dx^\mu}$$

Since the Legendre polynomials P_v(z) and P_v(x) don't differ by overall imaginary factors (EQ 8.6.18, Rodrigues' formula), it would seem that one could write
$$P^\mu_\nu(z)=\frac{P^\mu_\nu(x)}{i\,^\mu}$$

However, calculating the complex-argument polynomial from the real-argument polynomial this way gives numerically different values than using the complex formula directly. What am I missing in the relationship between these definitions? Thanks!

 

[avikarto]

Some more info for clarification - from different sources, these expressions can be written (for $\nu=\mu=m$) as...

According to Abramowitz & Stegun, EQ 8.6.6,
$$P^m_m(z)=\frac{(z^2-1)^{m/2}}{2^m m!}\cdot\frac{d^{2m} (z^2-1)^m}{dz^{2m}}$$
According to Arfken 85, Section 12.5,
$$P^m_m(z)=\frac{(1-z^2)^{m/2}}{2^m m!}\cdot\frac{d^{2m} (z^2-1)^m}{dz^{2m}}$$
According to Wolfram MathWorld,
$$P^m_m(z)=(-1)^m\frac{(1-z^2)^{m/2}}{2^m m!}\cdot\frac{d^{2m} (z^2-1)^m}{dz^{2m}}$$

Testing some calculations arbitrarily for m=3, these come out to be...

Abram: $P^m_m(Cos(z))=-15\,i\,Sin^3(z)$
Arfken: $P^m_m(Cos(z))=15\,Sin^3(z)$
Wolfram: $P^m_m(Cos(z))=-15\,Sin^3(z)$
(note, Wolfram's appears to be real valued. Dividing by $i^m$ as in the OP would bring this one in line with Abramowitz.)

Clearly, all of these can't simultaneously be right. Something about the general state of the definition for $P^m_m(z)$ seems to have serious issues. Does anyone know what is going on here?

Supposedly, the $(-1)^m$ in Wolfram's definition is a phase which makes it differ from Arfken, but why is a phase included at all in something that is purely mathematical and not at all physical? Aren't the polynomials just solutions to a mathematical equation with no physical meaning?