Askey scheme of hypergeometric orthogonal polynomials
E697762
The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Askey scheme of hypergeometric orthogonal polynomials canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7871835 — 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: Askey scheme of hypergeometric orthogonal polynomials Context triple: [Jacobi polynomials, belongsTo, Askey scheme of hypergeometric orthogonal polynomials]
-
A.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
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B.
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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C.
Askey–Wilson algebra
The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
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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.
Selberg integral
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
- 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: Askey scheme of hypergeometric orthogonal polynomials Target entity description: The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
-
A.
Orthogonal Polynomials
Orthogonal Polynomials is a classic mathematical monograph by Gábor Szegő that systematically develops the theory and applications of orthogonal polynomial systems in analysis and approximation theory.
-
B.
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.
-
C.
Askey–Wilson algebra
The Askey–Wilson algebra is a quadratic algebra arising in the theory of orthogonal polynomials and quantum groups, closely linked to the Askey–Wilson polynomials and related integrable models.
-
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.
Selberg integral
The Selberg integral is a fundamental multidimensional generalization of Euler’s beta integral that plays a central role in random matrix theory, combinatorics, and special functions.
- F. None of above. chosen
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
hierarchical scheme
ⓘ
mathematical classification scheme ⓘ taxonomy of orthogonal polynomials ⓘ |
| basedOn | limit relations between families of orthogonal polynomials ⓘ |
| characterizedBy |
hypergeometric or basic hypergeometric representations
ⓘ
orthogonality relations ⓘ three-term recurrence relations ⓘ |
| developedBy | James Wilson NERFINISHED ⓘ |
| extendedTo | q-Askey scheme NERFINISHED ⓘ |
| field |
basic hypergeometric functions
ⓘ
hypergeometric functions ⓘ orthogonal polynomials ⓘ special functions ⓘ |
| includesFamily |
Al-Salam–Chihara polynomials
NERFINISHED
ⓘ
Askey–Wilson polynomials NERFINISHED ⓘ Charlier polynomials NERFINISHED ⓘ Gegenbauer polynomials NERFINISHED ⓘ Hahn polynomials NERFINISHED ⓘ Hermite polynomials NERFINISHED ⓘ Jacobi polynomials NERFINISHED ⓘ Krawtchouk polynomials NERFINISHED ⓘ Laguerre polynomials NERFINISHED ⓘ Meixner polynomials NERFINISHED ⓘ Meixner–Pollaczek polynomials NERFINISHED ⓘ Racah polynomials NERFINISHED ⓘ Racah polynomials on finite sets ⓘ Wilson polynomials NERFINISHED ⓘ big q-Jacobi polynomials NERFINISHED ⓘ big q-Laguerre polynomials NERFINISHED ⓘ continuous Hahn polynomials NERFINISHED ⓘ continuous dual Hahn polynomials ⓘ continuous q-Jacobi polynomials NERFINISHED ⓘ continuous q-ultraspherical polynomials NERFINISHED ⓘ discrete q-Hermite polynomials NERFINISHED ⓘ dual Hahn polynomials NERFINISHED ⓘ dual q-Hahn polynomials ⓘ little q-Jacobi polynomials NERFINISHED ⓘ little q-Laguerre polynomials NERFINISHED ⓘ q-Bessel polynomials NERFINISHED ⓘ q-Charlier polynomials NERFINISHED ⓘ q-Hahn polynomials NERFINISHED ⓘ q-Krawtchouk polynomials NERFINISHED ⓘ q-Laguerre polynomials NERFINISHED ⓘ q-Meixner polynomials NERFINISHED ⓘ q-Racah polynomials NERFINISHED ⓘ |
| introducedBy | Richard Askey NERFINISHED ⓘ |
| organizedBy | degeneration limits of parameters ⓘ |
| organizes |
basic hypergeometric orthogonal polynomials
ⓘ
hypergeometric orthogonal polynomials ⓘ |
| topFamily |
Racah polynomials
NERFINISHED
ⓘ
Wilson polynomials NERFINISHED ⓘ |
| usedIn |
approximation theory
ⓘ
mathematical physics ⓘ spectral theory of operators ⓘ |
How these facts were elicited
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Input
Subject: Askey scheme of hypergeometric orthogonal polynomials Description of subject: The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.