# Conventions in defining spherical harmonics and associated Legendre polynomials

## Relevant Background

Spherical harmonics are defined with several different conventions: the definition used in quantum mechanics according to Wikipedia is

$Y_l^{\,m}(\theta,\phi) = (-1)^m\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_l^{\,m}(\cos\theta)\,e^{im\phi}$, where $P_l^{\,m}(x) = (-1)^m\ (1-x^2)^{m/2}\ \frac{d^m}{dx^m}P_l(x)$, for $m\geq 0$, and $P_l^{\,-m} = (-1)^m\frac{(l-m)!}{(l+m)!}P_l^{\,m}$.

The NIST definition as seen in the NIST Digital Library of Mathematical Functions (NIST DLMF) is

$Y_l^{\,m}(\theta,\phi) = \sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_l^{\,m}(\cos\theta)\,e^{im\phi}$, with the same definition for $P_l^{\,m}(x)$ as Wikipedia.

On the other hand, Griffiths' definition in Introduction to Quantum Mechanics is

$Y_l^{\,m}(\theta,\phi) = \epsilon\sqrt{\frac{2l+1}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}P_l^{\,m}(\cos\theta)\,e^{im\phi}$, where $\epsilon \equiv \begin{cases}(-1)^m &m\geq0\\ 1&m\leq 0 \end{cases}$ and $P_l^{\,m}=(1-x^2)^{|m|/2}\frac{d^{|m|}}{dx^{|m|}}P_l(x)$ for and $|m| \leq l$.

In all three cases the Legendre polynomials are defined as $P_0(x) = 0$, $P_1(x)=x,$ $P_2(x)=\frac{1}{2}(3x^2-1)$, i.e. via $\sum_{n=0}^\infty P_n(x)\,h^n = (1-2xh+h^2)^{-1/2}$.

The Wikipedia and NIST definitions differ only by $(-1)^m$. Griffiths' defines $P_l^{\,m}$ to be symmetric under $m \leftrightarrow -m$. One can check that this makes Griffiths' and Wikipedia's $P_l^{\,m}$ agree for $m >0$ and $m$ even. For $m>0$ and $m$ odd, Griffiths' $P_l^{\,m}$ disagrees with Wikipedia's by a minus sign.

Clearly, Griffiths' definitions are nice because $P_l^{\,-m} = P_l^{\,m}$ meaning that in $Y_l^{\,m}$ the only parts that depend on the sign of $m$ are $\epsilon$ and $e^{im\phi}$. Griffiths' points out in a footnote that $\epsilon$ is included to be consistent with the notation he uses for angular momentum.

## Questions

(1) Is one definition of $P_l^{\,m}$ more "natural" than the other? Is a particular definition more prevalent in a particular subfield?

(2) What is the reason for including the extra phase of $(-1)^m$ in $Y_l^{\,m}$ in the Wikipedia definition.

(3) What exactly is the point of $\epsilon$ in Griffiths' definition of $Y_l^{\,m}$? What is the connection to angular momentum?

Thanks in advance! I just joined; this is my first post.