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Darboux transforms on Band Matrices, Weights and associated Polynomials | | Mark Adler
; Pierre van Moerbeke
; | | Date: |
27 Oct 2000 | | Subject: | Exactly Solvable and Integrable Systems; Classical Analysis and ODEs; Mathematical Physics | nlin.SI math-ph math.CA math.MP | | Abstract: | Classically, it is well known that a single weight on a real interval leads to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP and Riemann-Hilbert problems", Comm. Math. Phys. 207, pp. 589-620 (1999), we have shown that $m$-periodic sequences of weights lead to "moments", polynomials defined by determinants of matrices involving these moments and $2m+1$-step relations between them, thus leading to $2m+1$-band matrices $L$. Given a Darboux transformations on $L$, which effect does it have on the $m$-periodic sequence of weights and on the associated polynomials ? These questions will receive a precise answer in this paper. The methods are based on introducing time parameters in the weights, making the band matrix $L$ evolve according to the so-called discrete KP hierarchy. Darboux transformations on that $L$ translate into vertex operators acting on the $ au$-function. | | Source: | arXiv, nlin.SI/0010048 | | Services: | Forum | Review | PDF | Favorites | |
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