3.7 Fully normalized associated Legendre functions

For convenience, let us introduce the abbreviations $t=\sin\bar{\varphi}$ and $u=\cos\bar{\varphi}$. The fully normalized associated Legendre functions $\bar{P}_{mn}(t)$, sometimes also called fully normalized harmonics, can be computed from the conventional associated Legendre functions $P_{n,m}$ by (Torge, 1991):

$$\begin{split}\bar{P}_{n,m}(t) = \sqrt{k(2n+1)\frac{(n-m)!}{(n... ...t{for $m = 0$}\ 2&\text{for $m\ne 0$}. \end{cases} \end{split}$$ (25)

The associated Legendre functions can be computed with the following recursive formulas (e.g., Bronstein and Semendjajew, 1991, Abramowitz and Stegun, 1972):

$$\begin{split}P_{n+1,0}(t)&=(2n+1) t P_{n,0}(t) - n P_{n-1,... ..._{n,m}(t)&=P_{n-2,m}(t) + (2n-1) u P_{n-1,m-1}(t) \end{split}$$ (26)

with the starting values

$$P_{0,0}(t) =1$$

$$P_{1,0}(t) =t, \quad P_{1,1}(t) =u$$

$$P_{2,0}(t) =\frac{3}{2} t^{2}-\frac{1}{2}, \quad P_{2,1}(t) =3 u t, \quad P_{2,2}(t) =3 u^{2}.$$

However, these recursion formulas become numerically unstable for large $n$ and $m$ ($>120$) and you may have to use other, more sophisticated formulas. These can be found in, for example, Paul (1978) and Holmes and Featherstone (2002). Here, we reproduce one method from Holmes and Featherstone (2002) for convenience:

For the fully normalized non-sectorial (i.e., $n>m$) $\bar{P}_{n,m}(t)$ you can use the following recursion:

$$\bar{P}_{n,m}(t) =a_{n,m} t \bar{P}_{n-1,m}(t) - b_{n,m} \bar{P}_{n-2,m}(t), n>m$$ (27)
where

$$a_{n,m} = \sqrt{\frac{(2n-1)(2n+1)}{(n-m)(n+m)}},$$

$$b_{n,m} = \sqrt{\frac{(2n+1)(n+m-1)(n-m-1)}{(n-m)(n+m)(2n-3)}}.$$

The sectorial (i.e. $n=m$) $\bar{P}_{m,m}(t)$ serve as seed values for the recursion in (27). Starting from $\bar{P}_{0,0}(t)=1$ and $\bar{P}_{1,1}(t)=\sqrt{3} u$, they can be computed from

$$\bar{P}_{m,m}(t) =u \sqrt{\frac{2m+1}{2m}} \bar{P}_{m-1,m-1}(t), m>1 ,$$ (28)
such that

$$\bar{P}_{m,m}(t) =u^{m} \sqrt{3}\prod_{i=2}^{m}\sqrt{\frac{2i+1}{2i}}, m>1 .$$ (29)