# Legendre polynomials

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##### 1: 18.3 Definitions
###### §18.3 Definitions
For exact values of the coefficients of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$, the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$, the Chebyshev polynomials $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, the Legendre polynomials $P_{n}\left(x\right)$, the Laguerre polynomials $L_{n}\left(x\right)$, and the Hermite polynomials $H_{n}\left(x\right)$, see Abramowitz and Stegun (1964, pp. 793–801). … Legendre polynomials are special cases of Legendre functions, Ferrers functions, and associated Legendre functions (§14.7(i)). …
##### 2: 18.30 Associated OP’s
###### Associated LegendrePolynomials
18.30.6 $P_{n}\left(x;c\right)=P^{(0,0)}_{n}\left(x;c\right),$ $n=0,1,\dots$.
(These polynomials are not to be confused with associated Legendre functions §14.3(ii).) For further results on associated Legendre polynomials see Chihara (1978, Chapter VI, §12); on associated Jacobi polynomials, see Wimp (1987) and Ismail and Masson (1991). …
##### 3: 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 $P_{n}\left(x\right)=\frac{1}{2^{n}n!}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}% }\left(x^{2}-1\right)^{n},$
##### 4: 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). …
##### 5: 18.4 Graphics Figure 18.4.4: Legendre polynomials P n ⁡ ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
##### 7: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. …
##### 8: 18.5 Explicit Representations
$P_{0}\left(x\right)=1,$
$P_{1}\left(x\right)=x,$
$P_{2}\left(x\right)=\tfrac{3}{2}x^{2}-\tfrac{1}{2},$
$P_{3}\left(x\right)=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x,$
##### 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 $P_{n}\left(\cos\theta\right)=\frac{1}{\pi}\int_{0}^{\pi}(\cos\theta+i\sin% \theta\cos\phi)^{n}\mathrm{d}\phi.$