Legendre functions can be of first and second *kinds*; $P$, $Q$.

They can have *order* $\mu$ and *degree* $\nu$; $P^\mu_\nu$ $Q^\mu_\nu$

But do they also have *Types*? Some of the numerical software I am using defines "types" for Legendre. In fact, most if not all seem to.

Mathematica defines "types" 2 and 3

http://functions.wolfram.com/HypergeometricFunctions/LegendreP2General/02/ http://functions.wolfram.com/HypergeometricFunctions/LegendreP3General/02/

And the MPmath software I am using(via Sagemath) does the same.

http://mpmath.googlecode.com/svn/trunk/doc/build/functions/orthogonal.html#legenp

The types do not agree for non-integer orders $\mu$. Confusingly, there does not appear to be a type 1 in either source.

So my question is: What are these types? Are they just numerical conventions, or are they important mathematical subtleties which can affect the solutions to Legendre's differential equation? (I'm particularly worried about the behaviour of the exponent m/2 given in the formulae for irrational m on negative numbers.)