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Stationary reduction method based on nonisospectral deformation of orthogonal polynomials, and discrete Painlev\'{e}-type equations
- Publication Year :
- 2024
-
Abstract
- In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlev\'{e}-type (d-P-type) equations. We successfully apply this approach to (bi)orthogonal polynomials satisfying ordinary orthogonality, $(1,m)$-biorthogonality, generalized Laurent biorthogonality, Cauchy biorthogonality and partial-skew orthogonality. As a result, several seemingly novel classes of high order d-P-type equations, along with their particular solutions and respective Lax pairs, are derived. Notably, the d-P-type equation related to the Cauchy biorthogonality can be viewed as a stationary reduction of a nonisospectral generalization involving the first two flows of the Toda hierarchy of CKP type. Additionally, the d-P-type equation related to the partial-skew orthogonality is associated with the nonisospectral Toda hierarchy of BKP type, and this equation is found to admit a solution expressed in terms of Pfaffians.<br />Comment: 88 pages; Several typos are corrected
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2407.09875
- Document Type :
- Working Paper