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

Xiang Ke Chang; Xiao Min Chen; Xing Biao Hu; Hon Wah Tam

doi:10.1088/1751-8113/48/1/015204

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

Xiang Ke Chang, Xiao Min Chen, Xing Biao Hu, Hon Wah Tam

Department of Computer Science

Research output: Contribution to journal › Journal article › peer-review

17 Citations (Scopus)

Abstract

By introducing some special bi-orthogonal polynomials, we derive the so-called 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-quotient-difference (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 quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

Original language	English
Article number	015204
Journal	Journal of Physics A: Mathematical and Theoretical
Volume	48
Issue number	1
DOIs	https://doi.org/10.1088/1751-8113/48/1/015204
Publication status	Published - 9 Jan 2015

Scopus Subject Areas

Statistical and Nonlinear Physics
Statistics and Probability
Modelling and Simulation
Mathematical Physics
Physics and Astronomy(all)

User-Defined Keywords

discrete integrable systems
Lax pair
orthogonal polynomials

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10.1088/1751-8113/48/1/015204

Cite this

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title = "About several classes of bi-orthogonal polynomials and discrete integrable systems",

abstract = "By introducing some special bi-orthogonal polynomials, we derive the so-called 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-quotient-difference (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 quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.",

keywords = "discrete integrable systems, Lax pair, orthogonal polynomials",

author = "Chang, {Xiang Ke} and Chen, {Xiao Min} and Hu, {Xing Biao} and Tam, {Hon Wah}",

note = "This work was partially supported by the National Natural Science Foundation of China (Grant no. 11331008, 11371251), the Knowledge Innovation Program of LSEC, the Institute of Computational Mathematics, AMSS, CAS, and the Hong Kong Research Grants Council (Grant no. GRF HKBU 202512).",

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AU - Chang, Xiang Ke

AU - Chen, Xiao Min

AU - Hu, Xing Biao

AU - Tam, Hon Wah

N1 - This work was partially supported by the National Natural Science Foundation of China (Grant no. 11331008, 11371251), the Knowledge Innovation Program of LSEC, the Institute of Computational Mathematics, AMSS, CAS, and the Hong Kong Research Grants Council (Grant no. GRF HKBU 202512).

PY - 2015/1/9

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N2 - By introducing some special bi-orthogonal polynomials, we derive the so-called 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-quotient-difference (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 quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

AB - By introducing some special bi-orthogonal polynomials, we derive the so-called 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-quotient-difference (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 quotient-quotient-difference (hQQD) scheme, respectively. In addition, the corresponding Lax pairs are provided.

KW - discrete integrable systems

KW - Lax pair

KW - orthogonal polynomials

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JO - Journal of Physics A: Mathematical and Theoretical

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

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