# Legendre polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
For expressions of ultraspherical, Chebyshev, and Legendre polynomials in terms of Jacobi polynomials, see §18.7(i). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
##### 2: 18.41 Tables
For $P_{n}\left(x\right)$ ($=\mathsf{P}_{n}\left(x\right)$) see §14.33. … For $P_{n}\left(x\right)$, $L_{n}\left(x\right)$, and $H_{n}\left(x\right)$ see §3.5(v). …
##### 3: 10.59 Integrals
10.59.1 $\int_{-\infty}^{\infty}e^{ibt}\mathsf{j}_{n}\left(t\right)\,\mathrm{d}t=\begin% {cases}\pi i^{n}P_{n}\left(b\right),&-11,\end{cases}$
where $P_{n}$ is the Legendre polynomial18.3). …
##### 4: 10.60 Sums
Then with $P_{n}$ again denoting the Legendre polynomial of degree $n$,
10.60.1 $\frac{\cos w}{w}=-\sum_{n=0}^{\infty}(2n+1)\mathsf{j}_{n}\left(v\right)\mathsf% {y}_{n}\left(u\right)P_{n}\left(\cos\alpha\right),$ $|ve^{\pm i\alpha}|<|u|$.
10.60.7 $e^{iz\cos\alpha}=\sum_{n=0}^{\infty}(2n+1)i^{n}\mathsf{j}_{n}\left(z\right)P_{% n}\left(\cos\alpha\right),$
10.60.8 $e^{z\cos\alpha}=\sum_{n=0}^{\infty}(2n+1){\mathsf{i}^{(1)}_{n}}\left(z\right)P% _{n}\left(\cos\alpha\right),$
##### 5: 14.7 Integer Degree and Order
###### §14.7(i) $\mu=0$
where $P_{n}\left(x\right)$ is the Legendre polynomial of degree $n$. For additional properties of $P_{n}\left(x\right)$ see Chapter 18. …
14.7.4 $W_{n-1}(x)=\sum_{k=1}^{n}\frac{1}{k}P_{k-1}\left(x\right)P_{n-k}\left(x\right).$
14.7.13 $P_{n}\left(x\right)=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}% }\left(x^{2}-1\right)^{n},$
##### 6: 10.54 Integral Representations
10.54.2 $\mathsf{j}_{n}\left(z\right)=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P% _{n}\left(\cos\theta\right)\sin\theta\,\mathrm{d}\theta.$
10.54.3 $\mathsf{k}_{n}\left(z\right)=\frac{\pi}{2}\int_{1}^{\infty}e^{-zt}P_{n}\left(t% \right)\,\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi.$
For the Legendre polynomial $P_{n}$ and the associated Legendre function $Q_{n}$ see §§18.3 and 14.21(i), with $\mu=0$ and $\nu=n$. …
##### 8: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. …
##### 9: 18.10 Integral Representations
###### Legendre
18.10.2 $P_{n}\left(\cos\theta\right)=\frac{2^{\frac{1}{2}}}{\pi}\int_{0}^{\theta}\frac% {\cos\left((n+\tfrac{1}{2})\phi\right)}{(\cos\phi-\cos\theta)^{\frac{1}{2}}}\,% \mathrm{d}\phi,$ $0<\theta<\pi$.
18.10.5 $P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin% \theta\cos\phi)^{n}\,\mathrm{d}\phi.$