A generalization of Laurent biorthogonal polynomials and related integrable lattices

Wang, Bao; Chang, Xiang-Ke; Yue, Xiaolu

doi:10.1088/1751-8121/ac6405

Cited by 3 publications

(3 citation statements)

References 41 publications

(35 reference statements)

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“…As they stand the results reported in the above works do not translate directly into the ones we seek. Systems of bi-orthogonal Laurent polynomials [71] would be expected to provide an equivalent framework to the system we treat here, however we prefer our approach because of its direct linkage to the random matrix application described earlier.…”

Section: Motivationmentioning

confidence: 99%

“…Recurrence RelationsAt this point we turn to the recurrence n → n + 1 structures in the 2 j − k and j − 2k systems, focusing initially on the scalar forms. Throughout this section we primarily use the Dodgson condensation identity (2.2) in order to prove the following results, which furnish the tools that are capable of extension to the more general pj − qk systems.4 4 In[71] it is shown that the j − qk biorthogonal polynomials satisfy a (q + 2)-term recurrence relation, without explicit descriptions of the recurrence coefficients.…”

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confidence: 99%

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Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Gharakhloo

Witte

2023

Constr Approx

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We construct the systems of bi-orthogonal polynomials on the unit circle where the Toeplitz structure of the moment determinants is replaced by $$\det (w_{2j-k})_{0\le j,k \le N-1} $$ det ( w 2 j - k ) 0 ≤ j , k ≤ N - 1 and the corresponding Vandermonde modulus squared is replaced by $$\prod _{1 \le j < k \le N}(\zeta _k - \zeta _j)(\zeta ^{-2}_k - \zeta ^{-2}_j) $$ ∏ 1 ≤ j < k ≤ N ( ζ k - ζ j ) ( ζ k - 2 - ζ j - 2 ) . This is the simplest case of a general system of $$pj-qk$$ p j - q k with p, q co-prime integers. We derive analogues of the structures well known in the Toeplitz case: third order recurrence relations, determinantal and multiple-integral representations, their reproducing kernel and Christoffel–Darboux sum, and associated (Carathéodory) functions. We close by giving full explicit details for the system defined by the simple weight $$ w(\zeta )=e^{\zeta }$$ w ( ζ ) = e ζ , which is a specialisation of a weight arising from averages of moments of derivatives of characteristic polynomials over $$\textrm{USp}(2N)$$ USp ( 2 N ) , $$\textrm{SO}(2N)$$ SO ( 2 N ) and $$\textrm{O}^-(2N)$$ O - ( 2 N ) .

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Section: Motivationmentioning

confidence: 99%

mentioning

confidence: 99%

Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Gharakhloo

Witte

2023

Constr Approx

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“…The dynamics of the leapfrog map leads to the so-called relativistic Toda equation in the theory of classical integrable systems, which exhibits a potential connection to the Laurent bi-orthogonal polynomials [30]. Therefore, in section 4, we construct a non-commutative version of Laurent biorthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

School Mathematics Textbooks in China

Wang¹

2015

East China Normal University Scientific Reports

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Let µ and ν be two probability measures on R d , where µ(dx) = e −V (x) dx R d e −V (x) dx for some V ∈ C 1 (R d ). Explicit sufficient conditions on V and ν are presented such that µ * ν satisfies the log-Sobolev, Poincaré and super Poincaré inequalities. In particular, if V (x) = λ|x| 2 for some λ > 0 and ν(e λθ|•| 2 ) < ∞ for some θ > 1, then µ * ν satisfies the log-Sobolev inequality. This improves and extends the recent results on the log-Sobolev inequality derived in [20] for convolutions of the Gaussian measure and compactly supported probability measures. On the other hand, it is well known that the log-Sobolev inequality for µ * ν implies ν(e ε|•| 2 ) < ∞ for some ε > 0. RésuméDes conditions explicites suffisantes sur V et ν sont présentées telles que µ * ν satisfait des inégalités de Sobolev logarithmique, de Poincaré et de super-Poincaré. En particulier, si V (x) = λ|x| 2 pour quelque λ > 0 et ν(e λθ|•| 2 ) < ∞ avec θ > 1, alors µ * ν satisfait l'inégalité de Sobolev logarithmique. Cela améliore et étend des résultats récents sur l'inégalité de Sobolev logarithmique obtenus dans [20] pour des convolutions de la mesure de Gauss et des mesures de probabilité à support compact. D'autre part, il est bien connu que l'inégalité de Sobolev logarithmique pour µ * ν implique ν(e ε|•| 2 ) < ∞ pour quelque ε > 0.

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Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials

Ito

2025

Discrete Mathematics

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A generalization of Laurent biorthogonal polynomials and related integrable lattices

Cited by 3 publications

References 41 publications

Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

Modulated Bi-Orthogonal Polynomials on the Unit Circle: The $$2j-k$$ and $$j-2k$$ Systems

School Mathematics Textbooks in China

Generalized Schröder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials

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