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.
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
NERFINISHED
ⓘ
many identities in q-series ⓘ q-beta integral NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| limitAs q→1 | Selberg integral NERFINISHED ⓘ |
| namedAfter | Atle Selberg (via its classical analogue) NERFINISHED ⓘ |
| relatedTo |
Askey–Wilson integral
NERFINISHED
ⓘ
Jackson integral ⓘ Koornwinder polynomials NERFINISHED ⓘ Macdonald polynomials NERFINISHED ⓘ Selberg integral NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.