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1: 18.3 Definitions
§18.3 Definitions
Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, normalizations, leading coefficients, and parameter constraints. …
Name p n ( x ) ( a , b ) w ( x ) h n k n k ~ n / k n Constraints
Legendre P n ( x ) ( - 1 , 1 ) 1 2 / ( 2 n + 1 ) 2 n ( 1 2 ) n / n ! 0
Shifted Legendre P n * ( x ) ( 0 , 1 ) 1 1 / ( 2 n + 1 ) 2 2 n ( 1 2 ) n / n ! - 1 2 n
For exact values of the coefficients of the Jacobi polynomials P n ( α , β ) ( x ) , the ultraspherical polynomials C n ( λ ) ( x ) , the Chebyshev polynomials T n ( x ) and U n ( x ) , the Legendre polynomials P n ( x ) , the Laguerre polynomials L n ( x ) , and the Hermite polynomials H n ( x ) , 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 Legendre Polynomials
18.30.6 P n ( x ; c ) = P n ( 0 , 0 ) ( x ; c ) , n = 0 , 1 , .
18.30.7 P n ( x ; c ) = = 0 n c + c P ( x ) P n - ( x ) .
(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) μ = 0
where P n ( x ) is the Legendre polynomial of degree n . For additional properties of P n ( x ) see Chapter 18. …
14.7.4 W n - 1 ( x ) = k = 1 n 1 k P k - 1 ( x ) P n - k ( x ) .
14.7.13 P n ( x ) = 1 2 n n ! d n d x n ( x 2 - 1 ) n ,
4: 18.41 Tables
For P n ( x ) ( = P n ( x ) ) see §14.33. … For P n ( x ) , L n ( x ) , and H n ( x ) see §3.5(v). …
5: 18.4 Graphics
See accompanying text
Figure 18.4.4: Legendre polynomials P n ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify
6: 18.17 Integrals
Legendre
Legendre
Legendre
Legendre
Legendre
7: 18.6 Symmetry, Special Values, and Limits to Monomials
For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1. …
Table 18.6.1: Classical OP’s: symmetry and special values.
p n ( x ) p n ( - x ) p n ( 1 ) p 2 n ( 0 ) p 2 n + 1 ( 0 )
P n ( x ) ( - 1 ) n P n ( x ) 1 ( - 1 ) n ( 1 2 ) n / n ! 2 ( - 1 ) n ( 1 2 ) n + 1 / n !
8: 18.5 Explicit Representations
Table 18.5.1: Classical OP’s: Rodrigues formulas (18.5.5).
p n ( x ) F ( x ) κ n
P n ( x ) 1 - x 2 ( - 2 ) n n !
P 0 ( x ) = 1 ,
P 1 ( x ) = x ,
P 2 ( x ) = 3 2 x 2 - 1 2 ,
P 3 ( x ) = 5 2 x 3 - 3 2 x ,
9: 18.10 Integral Representations
Legendre
18.10.2 P n ( cos θ ) = 2 1 2 π 0 θ cos ( ( n + 1 2 ) ϕ ) ( cos ϕ - cos θ ) 1 2 d ϕ , 0 < θ < π .
18.10.5 P n ( cos θ ) = 1 π 0 π ( cos θ + i sin θ cos ϕ ) n d ϕ .
Table 18.10.1: Classical OP’s: contour integral representations (18.10.8).
p n ( x ) g 0 ( x ) g 1 ( z , x ) g 2 ( z , x ) c Conditions
P n ( x ) 1 z - 1 ( 1 - 2 x z + z 2 ) - 1 2 0
P n ( x ) 1 z 2 - 1 2 ( z - x ) 1 x
10: 18.8 Differential Equations
Table 18.8.1: Classical OP’s: differential equations A ( x ) f ′′ ( x ) + B ( x ) f ( x ) + C ( x ) f ( x ) + λ n f ( x ) = 0 .
f ( x ) A ( x ) B ( x ) C ( x ) λ n
P n ( x ) 1 - x 2 - 2 x 0 n ( n + 1 )