Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications

Komori, Yasushi

doi:10.3842/sigma.2009.054

Cited by 38 publications

(73 citation statements)

References 23 publications

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“…In the trigonometric (II) case, Corollary 4.3 recovers the result of Mimachi [Mim01] while the elliptic generalization of Mimachi's result was given in [KNS09].…”

Section: Special Casessupporting

confidence: 64%

“…In this paper we constructed a Chalykh-Feigin-Sergeev-Veselov type generalization of van Diejen's analytic difference operator and obtained kernel function identities for this operator and its various limiting cases. Using the kernel functions for the deformed Koornwinder-van Diejen type operators it is possible to construct eigenfunctions and eigenvalues of these operators using methods developed in, for example, [Mim01,KNS09,HL10]. In particular, we believe that it is possible to construct generalizations of the Koornwinder polynomials, similar to the generalizations of the Macdonald polynomials in [SV09b], and extending the results in [SV09a] to the q-difference case.…”

Section: Final Remarksmentioning

confidence: 89%

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Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

2014

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We consider generalizations of the BC-type relativistic Calogero-Moser-Sutherland models, comprising of the rational, trigonometric, hyperbolic, and elliptic cases, due to Koornwinder and van Diejen, and construct an explicit eigenfunction for these generalizations. In special cases, we find the various kernel function identities, and also a Chalykh-Feigin-Sergeev-Veselov type deformation of these operators and their corresponding kernel functions, which generalize the known kernel functions for the Koornwinder-van Diejen models.Then it is a straightforward check to see that φ(x; m, −m) in (13) satisfies both difference equations.When (m J , m K ) = (m, +1/λm) for m ∈ Λ, then (27a) and (27b), using (29), becomesimplifies the four cases and allows us to find the common solution φ(x; m, +1/λm) in (13).When (m J , m K ) = (m, −1/λm) for m ∈ λ, then (27a) and (27b), using (29), becomerespectively. Choosing (34) a(m) − a(−1/λm) = −iλmβ/2 − iβ/2m (∀m ∈ Λ) reduces the equations to δ=± φ(X J + δX K ∓ iβ/2m; m, −1/λm)φ(−X J + δX K ± iβ/2m; m, −1/λm) φ(X J + δX K ± iβ/2m; m, −1/λm)φ(−X J + δX K ∓ iβ/2m; m, −1/λm) = δ=± s(X J + δX K ± iλmβ/2)s(X J + δX K ∓ iλmβ/2 ± iβ/m) s(X J + δX K ∓ iλmβ/2)s(X J + δX K ± iλmβ/2 ∓ iβ/m) 1 2 and δ=± φ(X J + δX K ± iλmβ/2; m, −1/λm)φ(−X J + δX K ∓ iλmβ/2; m, −1/λm) φ(X J + δX K ∓ iλmβ/2; m, −1/λm)φ(−X J + δX K ± iλmβ/2) = δ=± s(X J + δX K ∓ iβ/2m)s(X J + δX K ± iβ/2m ∓ iλmβ) s(X J + δX K ± iβ/2m)s(X J + δX K ∓ iβ/2m ± iλmβ) 1 2 ψ(X J − εiβ/m J ; m J ) ψ(X J ; m J ) · δ=± N K =J

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“…In the trigonometric (II) case, Corollary 4.3 recovers the result of Mimachi [Mim01] while the elliptic generalization of Mimachi's result was given in [KNS09].…”

Section: Special Casessupporting

confidence: 64%

Section: Final Remarksmentioning

confidence: 89%

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

2014

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“…Moreover it is remarkable that the polynomials e i (z) also coincide with Okounkov's [26]. In a recent work [22], Komori, Noumi and Shiraishi encountered the polynomials (1.7) in a context very different from ours. This provides a further interpretation of their origin.…”

Section: Introductionmentioning

confidence: 85%

A determinant formula for a holonomic q-difference system associated with Jackson integrals of type BCn

Aomoto

Ito

2009

Advances in Mathematics

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A Jackson integral of type BC n is a multisum generalization of the very-well-poised-balanced 2r ψ 2r basic hypergeometric series. We state an explicit product formula for the determinant of a matrix with entries given by the BC n type Jackson integrals. In order to show this, we treat the determinant as a solution of a holonomic q-difference equation. In particular we give the q-difference equation explicitly as a two-term recurrence relation, which the determinant satisfies, by introducing a set of new symmetric polynomials via the symplectic Schur functions.

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“…The results were interpreted as certain summations over the sets of tableaux of types C n and D n . While using the same technique as in [5], but replacing the Cauchy type kernel function by Mimachi's dual-Cauchy type one (as to the kernel functions, see [9,15]), we can study an explicit formula for the Koornwinder polynomials with one column diagrams.…”

Section: Koornwinder Polynomials Pmentioning

confidence: 99%

Macdonald Polynomials of Type C<sub>n</sub> with One-Column Diagrams and Deformed Catalan Numbers

Hoshino¹,

Shiraishi²

2018

SIGMA

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We present an explicit formula for the transition matrix C from the type C n degeneration of the Koornwinder polynomials P (1 r ) (x | a, −a, c, −c | q, t) with one column diagrams, to the type C n monomial symmetric polynomials m (1 r ) (x). The entries of the matrix C enjoy a set of three term recursion relations, which can be regarded as a (a, c, t)deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type (C n , C n ) Macdonald polynomials P (x | q; q, q) to the Hall-Littlewood polynomials P (Cn,Cn) (1 r ) (x | t; 0, t).

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Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications

Abstract: Abstract. A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.

Cited by 38 publications

References 23 publications

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

A determinant formula for a holonomic q-difference system associated with Jackson integrals of type BCn

Macdonald Polynomials of Type C<sub>n</sub> with One-Column Diagrams and Deformed Catalan Numbers

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