q-Selberg integral
E865111
generalization of Selberg integral
mathematical concept
multivariate basic hypergeometric integral
object of special function theory
q-analogue
The q-Selberg integral is a q-analogue of the classical Selberg integral, expressing a multivariate basic hypergeometric integral that generalizes many important identities in q-series and special function theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| q-Selberg integral canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10462129 — 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: q-Selberg integral Context triple: [Selberg integral, hasVariant, q-Selberg integral]
-
A.
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.
-
B.
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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C.
Askey scheme of hypergeometric orthogonal polynomials
The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
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D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
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E.
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.
- 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: q-Selberg integral Target entity description: The q-Selberg integral is a q-analogue of the classical Selberg integral, expressing a multivariate basic hypergeometric integral that generalizes many important identities in q-series and special function theory.
-
A.
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.
-
B.
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.
-
C.
Askey scheme of hypergeometric orthogonal polynomials
The Askey scheme of hypergeometric orthogonal polynomials is a hierarchical classification of families of (basic) hypergeometric orthogonal polynomials, organized by limit relations between them.
-
D.
Rogers–Ramanujan-type identities
Rogers–Ramanujan-type identities are a class of deep q-series and partition identities generalizing the classical Rogers–Ramanujan identities, with rich connections to combinatorics, number theory, and modular forms.
-
E.
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.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
generalization of Selberg integral
ⓘ
mathematical concept ⓘ multivariate basic hypergeometric integral ⓘ object of special function theory ⓘ q-analogue ⓘ |
| appearsIn |
literature on Macdonald–Koornwinder theory
ⓘ
studies of q-deformations of classical integrals ⓘ theory of multivariate basic hypergeometric functions ⓘ |
| definedOver | region [0,1]^n or q-discretized variants ⓘ |
| dependsOn |
complex parameters α, β, γ (or analogous parameters)
ⓘ
dimension n ⓘ parameter q ⓘ |
| field |
analytic number theory
ⓘ
basic hypergeometric series ⓘ combinatorics ⓘ mathematics ⓘ q-series ⓘ representation theory ⓘ special functions ⓘ |
| generalizes |
Selberg integral
ⓘ
many identities in q-series ⓘ q-beta integral ⓘ |
| hasProperty |
multivariate
ⓘ
provides q-analogues of beta-type integrals ⓘ reduces to various known q-integrals in special cases ⓘ symmetric in integration variables (under suitable parameters) ⓘ |
| involves |
basic hypergeometric products
ⓘ
multiple integration over a q-lattice or unit cube ⓘ q-Pochhammer symbol ⓘ |
| limitAs q→1 | Selberg integral ⓘ |
| namedAfter |
Atle Selberg (via its classical analogue)
ⓘ
linked to:
Selberg trace formula
|
| relatedTo |
Askey–Wilson integral
ⓘ
Jackson integral ⓘ Koornwinder polynomials ⓘ Macdonald polynomials ⓘ Selberg integral ⓘ basic hypergeometric functions ⓘ q-beta integrals ⓘ |
| usedFor |
applications in representation theory of quantum groups
ⓘ
computing norms of Macdonald-type polynomials ⓘ deriving orthogonality relations for q-orthogonal polynomials ⓘ evaluating partition functions in solvable lattice models ⓘ proving identities in basic hypergeometric series ⓘ |
| weightFunction | product of powers and q-shifted factorials in the variables ⓘ |
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: q-Selberg integral Description of subject: The q-Selberg integral is a q-analogue of the classical Selberg integral, expressing a multivariate basic hypergeometric integral that generalizes many important identities in q-series and special function theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.