Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems

Miki, Hiroshi

doi:10.3842/sigma.2012.008

Cited by 9 publications

(22 citation statements)

References 25 publications

(42 reference statements)

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“…Now we are ready to give the representations of SOPs in terms of Pfaffians. The readers may refer [2,38] for more information.…”

Section: Appendix B Sops and Pfaff Latticesmentioning

confidence: 99%

“…Discrete counterpart of Pfaff lattice. By introducing the so-called skew-Christoffel transformation for SOPs, and its inverse transformation, Miki, Goda and Tsujimoto [38] obtained a discrete integrable system in 1+1 dimension, which they call a discrete counterpart of the Pfaff lattice. We restate their derivation as follows: Introduce a series of polynomial sequences {P t n (z)} ∞ n=0 (Here t ∈ N 0 is a discrete variable) iterated by P 0 n (z) := P n (z),…”

Section: B4mentioning

confidence: 99%

“…which is regarded as a discrete counterpart of the t 1 flow of the Pfaff lattice by the authors [38] by noticing that (B.11) may lead to the moment evolution in (B.9) if a suitable limit is taken.…”

Section: B4mentioning

confidence: 99%

“…As a product originating from the study of RMT, the skew-orthogonal polynomials (SOPs) have been extensively studied in the literature ( see, e.g. [1,2,12,18,21,[36][37][38] ). Later, the so-called Pfaff lattice [2] was proposed, whose tau-functions are Pfaffians and the wave vectors SOPs.…”

mentioning

confidence: 99%

“…Later, the so-called Pfaff lattice [2] was proposed, whose tau-functions are Pfaffians and the wave vectors SOPs. And, in [38], the authors produced full-discrete integrable systems by considering the discrete spectral transformations of SOPs.…”

mentioning

confidence: 99%

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Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Chang

et al. 2018

Commun. Math. Phys.

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Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the taufunctions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble.Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Padé approximants. This yields the first example regarding the connection between integrable lattices and vector Padé approximants, for which Hietarinta, Joshi and Nijhoff pointed out that " This field remains largely to be explored. " in the recent monograph [27, Section 4.4] .

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“…Now we are ready to give the representations of SOPs in terms of Pfaffians. The readers may refer [2,38] for more information.…”

Section: Appendix B Sops and Pfaff Latticesmentioning

confidence: 99%

Section: B4mentioning

confidence: 99%

Section: B4mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Chang

et al. 2018

Commun. Math. Phys.

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Laurent skew orthogonal polynomials and related symplectic matrices

Miki

2020

Journal of Approximation Theory

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About several classes of bi-orthogonal polynomials and discrete integrable systems

Chang¹,

Chen²,

Hu³

et al. 2014

J. Phys. A: Math. Theor.

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By introducing some special bi-orthogonal polynomials, we derive the socalled discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka-Volterra (QD-type dhLV) system, together with their Lax pairs. These two known equations can be regarded as extensions of the QD algorithm. When this idea is applied to a higher analogue of the discrete-time Toda (HADT) equation and the quotient-quotientdifference (QQD) scheme proposed by Spicer, Nijhoff and van der Kamp, two extended systems are constructed. We call these systems the hungry forms of the higher analogue discrete-time Toda (hHADT) equation and the quotientquotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

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Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems

Cited by 9 publications

References 25 publications

Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Laurent skew orthogonal polynomials and related symplectic matrices

About several classes of bi-orthogonal polynomials and discrete integrable systems

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