A generalization of Selberg’s beta integral

Gustafson, Robert A.

doi:10.1090/s0273-0979-1990-15852-5

Cited by 91 publications

(77 citation statements)

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“…), see [9]. From Stirling's formula it follows that the polynomials are integrable on (iR) n with respect to both ∆ and ∆ + .…”

Section: Orthogonality Relationsmentioning

confidence: 97%

Multivariable Wilson polynomials and degenerate Hecke algebras

Groenevelt

2009

Sel. Math. New Ser.

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Abstract. We study a rational version of the double affine Hecke algebra associated to the nonreduced affine root system of type (C ∨ n , Cn). A certain representation in terms of difference-reflection operators naturally leads to the definition of nonsymmetric versions of the multivariable Wilson polynomials. Using the degenerate Hecke algebra we derive several properties, such as orthogonality relations and quadratic norms, for the nonsymmetric and symmetric multivariable Wilson polynomials. Mathematics Subject Classification (2000). Primary 33C52, 33C80; Secondary 20C08.

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“…), see [9]. From Stirling's formula it follows that the polynomials are integrable on (iR) n with respect to both ∆ and ∆ + .…”

Section: Orthogonality Relationsmentioning

confidence: 97%

Multivariable Wilson polynomials and degenerate Hecke algebras

Groenevelt

2009

Sel. Math. New Ser.

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“…(t, t 2n−2−j abcd; q) (t j+1 , t j ab, t j ac, t j ad, t j bc, t j bd, t j cd; q) , which may be found in [8].…”

Section: An α-Generalizationmentioning

confidence: 99%

“…(t, t 2n−2−j t 0 t 1 t 2 t 3 ; 0) (t j+1 , t j t 0 t 1 , t j t 0 t 2 , t j t 0 t 3 , t j t 1 t 2 , t j t 1 t 3 , t j t 2 t 3 ; 0) , (3.4) where the explicit evaluation for the integral is a result of Gustafson [8].…”

Section: Background and Notationmentioning

confidence: 99%

Vanishing integrals for Hall–Littlewood polynomials

Venkateswaran

2012

Transformation Groups

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It is well known that if one integrates a Schur function indexed by a partition λ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of λ have even multiplicity (resp. all parts of λ are even). In a recent paper of Rains and Vazirani, Macdonald polynomial generalizations of these identities and several others were developed and proved using Hecke algebra techniques. However at q = 0 (the Hall-Littlewood level), these approaches do not work, although one can obtain the results by taking the appropriate limit. In this paper, we develop a direct approach for dealing with this special case. This technique allows us to prove some identities that were not amenable to the Hecke algebra approach. Moreover, we are able to generalize some of the identities by introducing extra parameters. This leads us to a finite-dimensional analog of a recent result of Warnaar, which uses the Rogers-Szegő polynomials to unify some existing summation type formulas for Hall-Littlewood functions.

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“…To see this use (2.8) and the fact oo mn = l/J2gk(k+l),2h2k(x"(<T))/(q;q)k k=0 to see that m" is a constant multiple of \/{-qe2in, -ge~2,*")oo , where x"(er) = sinh(J" . We also note that one can prove formula (7.3) directly by making a change of variable v = e~^ , write the resulting integral as S-oo /Jj+i > replace v by wqi , interchange integration (which is now on [q, 1]) and summation, then evaluate the sum using the 6^6 sum (3.17) [27]. It is not surprising that the eV6 sum is what is behind (7.3), since after all, the evaluation of the 6% sum is equivalent to (3.8) when dy/ in (1.18) is any extremal measure.…”

Section: Applicationsmentioning

confidence: 99%