Rodrigues formula
E697757
Rodrigues formula is a classical representation that expresses certain families of orthogonal polynomials, such as Jacobi polynomials, in terms of derivatives of weight functions.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical formula
ⓘ
representation of orthogonal polynomials ⓘ |
| appliesTo |
Gegenbauer polynomials
NERFINISHED
ⓘ
Hermite polynomials NERFINISHED ⓘ Jacobi polynomials NERFINISHED ⓘ Laguerre polynomials NERFINISHED ⓘ Legendre polynomials NERFINISHED ⓘ |
| associatedWith |
Gaussian quadrature
ⓘ
Sturm–Liouville problems NERFINISHED ⓘ moment problems ⓘ second-order linear differential equations ⓘ |
| characterizes | classical orthogonal polynomial families ⓘ |
| context |
approximation theory
ⓘ
classical theory of special functions ⓘ spectral methods in numerical analysis ⓘ |
| describes | orthogonal polynomials ⓘ |
| expresses | orthogonal polynomials in terms of derivatives of weight functions ⓘ |
| field |
classical analysis
ⓘ
mathematical analysis ⓘ orthogonal polynomials ⓘ special functions ⓘ |
| hasForm | P_n(x) = 1/(w(x)) * d^n/dx^n [w(x) \, \\phi_n(x)] for suitable weight w and function \\phi_n ⓘ |
| implies |
existence of orthogonality relations with respect to the weight
ⓘ
polynomials satisfy a second-order differential equation ⓘ |
| namedAfter | Olinde Rodrigues NERFINISHED ⓘ |
| property | gives explicit expression for polynomial coefficients via derivatives ⓘ |
| relatedTo |
generating functions of orthogonal polynomials
ⓘ
orthogonality with respect to a measure ⓘ three-term recurrence relations ⓘ |
| relates | orthogonal polynomials and their weight functions ⓘ |
| requires |
nonnegative weight function on an interval
ⓘ
sufficient differentiability of the weight function ⓘ |
| specialCase |
Rodrigues formula for Hermite polynomials
NERFINISHED
ⓘ
Rodrigues formula for Jacobi polynomials NERFINISHED ⓘ Rodrigues formula for Laguerre polynomials ⓘ Rodrigues formula for Legendre polynomials NERFINISHED ⓘ |
| timePeriod | 19th century ⓘ |
| usedFor |
computing explicit forms of orthogonal polynomials
ⓘ
deriving differential equations satisfied by orthogonal polynomials ⓘ deriving recurrence relations of orthogonal polynomials ⓘ proving orthogonality properties ⓘ |
| usedIn |
mathematical physics
ⓘ
quantum mechanics NERFINISHED ⓘ solutions of Schrödinger-type equations via orthogonal polynomials ⓘ |
| uses |
nth derivative of a function
ⓘ
weight function ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.