I have used Ambisonics audio for 10 years. I have a grasp of the maths on a trigonometric level and have spent the last six months studying Stroud's Advanced Engineering Mathematics and am learning spherical harmonics. I have a few rather basic questions about associated Legendre polynomials that I have been following in a text that was included in the following thread. Spherical harmonics for dummies
Firstly, what is the meaning of the line over $P_m(x)$ in the second equation?
Secondly, what is the significance of the apostrophes on m and l? Please excuse me for the rather simple questions but it seems that this knowledge is assumed.
1) $\overline{P_m(x)}$ means complex complement of $P_m(x)$.
2) Those are not apostrophes but instead primes... often used to indicate a variable with its common associated name, but different from the general use.
For instance, a coordinate system might be $x^\prime, y^\prime, z^\prime$ to show it isn't the "usual" $x, y, z$, yet remind you of its source notation. Thus $l^\prime$ and $n^\prime$ are similar to $l$ and $n$, but could have values other than the $l$ and $n$ used elsewhere.
Clear?