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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Rodrigues formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871807 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Rodrigues formula Context triple: [Jacobi polynomials, satisfies, Rodrigues formula]
-
A.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
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B.
Hermite
Hermite is a French surname most famously associated with the 19th-century mathematician Charles Hermite, known for his contributions to number theory, algebra, and analysis.
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C.
Gauss multiplication formula
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
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D.
Gauss hypergeometric function
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
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E.
Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Rodrigues formula Target entity description: Rodrigues formula is a classical representation that expresses certain families of orthogonal polynomials, such as Jacobi polynomials, in terms of derivatives of weight functions.
-
A.
Jacobi polynomials
Jacobi polynomials are a family of classical orthogonal polynomials depending on two parameters, widely used in approximation theory, numerical analysis, and solutions of differential equations.
-
B.
Hermite
Hermite is a French surname most famously associated with the 19th-century mathematician Charles Hermite, known for his contributions to number theory, algebra, and analysis.
-
C.
Gauss multiplication formula
The Gauss multiplication formula is a classical identity in complex analysis that expresses the gamma function of a multiple of a variable as a product of gamma functions evaluated at shifted fractions of that variable.
-
D.
Gauss hypergeometric function
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
-
E.
Bernoulli polynomials
Bernoulli polynomials are a sequence of polynomials deeply connected to number theory and analysis, appearing in the study of special functions, series expansions, and the evaluation of sums of powers of integers.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Rodrigues formula Description of subject: Rodrigues formula is a classical representation that expresses certain families of orthogonal polynomials, such as Jacobi polynomials, in terms of derivatives of weight functions.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.