Statements (35)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
orthogonal polynomials
|
| gptkbp:category |
gptkb:software
|
| gptkbp:definedIn |
gptkb:Rodrigues'_formula
|
| gptkbp:degree |
n
|
| gptkbp:differential |
(1-x^2)y'' - 2xy' + n(n+1)y = 0
|
| gptkbp:field |
gptkb:mathematics
mathematical physics |
| gptkbp:firstFew |
P_0(x) = 1
P_1(x) = x P_2(x) = (3x^2 - 1)/2 P_3(x) = (5x^3 - 3x)/2 |
| gptkbp:generatingFunction |
(1-2xt+t^2)^{-1/2} = \\sum_{n=0}^\\infty P_n(x)t^n
|
| https://www.w3.org/2000/01/rdf-schema#label |
Legendre polynomial
|
| gptkbp:namedAfter |
gptkb:Adrien-Marie_Legendre
|
| gptkbp:orthogonalityRelation |
\\int_{-1}^1 P_m(x)P_n(x)dx = 2/(2n+1) \\delta_{mn}
|
| gptkbp:orthogonalOn |
interval [-1, 1]
|
| gptkbp:par |
P_n(-x) = (-1)^n P_n(x)
|
| gptkbp:real |
yes
|
| gptkbp:recurrence |
(n+1)P_{n+1}(x) = (2n+1)xP_n(x) - nP_{n-1}(x)
|
| gptkbp:relatedTo |
gptkb:Gegenbauer_polynomials
gptkb:Jacobi_polynomials gptkb:Chebyshev_polynomials associated Legendre polynomials |
| gptkbp:RodriguesFormula |
P_n(x) = (1/2^n n!) d^n/dx^n (x^2-1)^n
|
| gptkbp:roots |
all real and in (-1,1)
|
| gptkbp:satisfies |
gptkb:Legendre_differential_equation
|
| gptkbp:symbol |
P_n(x)
|
| gptkbp:type |
classical polynomial
|
| gptkbp:usedIn |
numerical analysis
approximation theory spherical harmonics solution of differential equations Gauss-Legendre quadrature |
| gptkbp:bfsParent |
gptkb:Hermite_polynomial
|
| gptkbp:bfsLayer |
7
|