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 (4)
| Label | Occurrences |
|---|---|
| Askey scheme of hypergeometric orthogonal polynomials canonical | 3 |
| q-Askey scheme | 2 |
| continuous q-ultraspherical polynomials | 1 |
| discrete q-Hermite polynomials | 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 ⓘ |
| extendedTo | q-Askey scheme ⓘ |
| field |
basic hypergeometric functions
ⓘ
hypergeometric functions ⓘ orthogonal polynomials ⓘ special functions ⓘ |
| includesFamily |
Al-Salam–Chihara polynomials
ⓘ
linked to:
Askey–Wilson polynomials
Askey–Wilson polynomials ⓘ Charlier polynomials ⓘ Gegenbauer polynomials ⓘ Hahn polynomials ⓘ Hermite polynomials ⓘ Jacobi polynomials ⓘ Krawtchouk polynomials ⓘ Laguerre polynomials ⓘ Meixner polynomials ⓘ Meixner–Pollaczek polynomials ⓘ Racah polynomials ⓘ Racah polynomials on finite sets ⓘ Wilson polynomials ⓘ big q-Jacobi polynomials ⓘ
linked to:
Askey–Wilson polynomials
big q-Laguerre polynomials ⓘ continuous Hahn polynomials ⓘ continuous dual Hahn polynomials ⓘ continuous q-Jacobi polynomials ⓘ continuous q-ultraspherical polynomials ⓘ discrete q-Hermite polynomials ⓘ dual Hahn polynomials ⓘ dual q-Hahn polynomials ⓘ little q-Jacobi polynomials ⓘ little q-Laguerre polynomials ⓘ q-Bessel polynomials ⓘ q-Charlier polynomials ⓘ q-Hahn polynomials ⓘ q-Krawtchouk polynomials ⓘ
linked to:
Krawtchouk polynomials
q-Laguerre polynomials ⓘ
linked to:
little q-Laguerre polynomials
q-Meixner polynomials ⓘ
linked to:
Askey–Wilson polynomials
q-Racah polynomials ⓘ
linked to:
Askey–Wilson polynomials
|
| introducedBy | Richard Askey ⓘ |
| organizedBy | degeneration limits of parameters ⓘ |
| organizes |
basic hypergeometric orthogonal polynomials
ⓘ
hypergeometric orthogonal polynomials ⓘ |
| topFamily |
Racah polynomials
ⓘ
Wilson polynomials ⓘ |
| usedIn |
approximation theory
ⓘ
mathematical physics ⓘ spectral theory of operators ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Askey scheme of hypergeometric orthogonal polynomials
Chebyshev polynomials of the first kind
→
belongsTo
→
Askey scheme of hypergeometric orthogonal polynomials
ⓘ
Askey scheme of hypergeometric orthogonal polynomials
→
includesFamily
→
discrete q-Hermite polynomials
ⓘ
linked to: Askey scheme of hypergeometric orthogonal polynomials
Askey scheme of hypergeometric orthogonal polynomials
→
includesFamily
→
continuous q-ultraspherical polynomials
ⓘ
linked to: Askey scheme of hypergeometric orthogonal polynomials
linked to: Askey scheme of hypergeometric orthogonal polynomials