Legendre polynomials and associated Legendre functions

The Legendre polynomials $P_k : [-1,1]\rightarrow \ensuremath{\mathbb{R}}$ , $k\ge 0$ , as classical orthogonal polynomials are given by their corresponding Rodrigues formula

$$P_k(x) := \frac{1}{2^k k!} \frac{\text{d}^k}{\text{d} x^k} \left(x^2-1\right)^k.$$

The associated Legendre functions $P_k^n : [-1,1] \rightarrow \ensuremath{\mathbb{R}}$ , $k \ge n \ge 0$ are defined by

$$P_k^n(x) := \left(\frac{(k-n)!}{(k+n)!}\right)^{1/2} \left(1-x^2\right)^{n/2} \frac{\text{d}^n}{\text{d} x^n} P_k(x).$$

For $n = 0$ , they coincide with the Legendre polynomials $P_k = P_k^0$ . The associated Legendre functions $P_{k}^n$ obey the three-term recurrence relation

$$P_{k+1}^n(x) = \frac{2k+1}{((k-n+1)(k+n+1))^{1/2}} x P_k^n(x) - \frac{((k-n)(k+n))^{1/2}}{((k-n+1)(k+n+1))^{1/2}} P_{k-1}^n(x)$$

for $k \ge n \ge 0$ , $P_{n-1}^n(x) = 0$ , $P_{n}^n(x) = \frac{\sqrt{(2n)!}}{2^n n!}\left(1-x^2\right)^{n/2}$ . For fixed $n$ , the set $\{P_k^n:\: k \ge n\}$ forms a set of orthogonal functions, i.e.,

$$\left\langle P_k^n,P_l^n \right\rangle = \int_{-1}^{1} P_k^n(x) P_l^n(x) x = \frac{2}{2k+1} \delta_{k,l}.$$

Again, we denote by $\bar{P}_{k}^n = \sqrt{\frac{2k+1}{2}} P_k^n$ the orthonormal associated Legendre functions. In the following, we allow also for $n < 0$ and set $P_{k}^n$ := $P_{k}^{-n}$ in this case.