Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

Zhedanov, Alexei

doi:10.1007/s11139-013-9515-x

Cited by 3 publications

(1 citation statement)

References 35 publications

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“…In particular, for the modification by e −xt we can understand the orthogonal polynomials for the measure e −xt dµ(x), where the measure µ corresponds to the orthogonality measure for a family of orthogonal polynomials in the Askey-scheme. By the work of Flaschka, Moser and others this is related to the Lax pair for the Toda lattice, see, e.g., [5,Section 4.6], [6,Section 2], [14,Section 2.8], [27,Section 2]. The recurrence coefficients for the monic orthogonal polynomials xp n (x; t) = p n+1 (x; t) + b n (t)p n (x; t) + c n (t)p n−1 (x; t) for the modified weight satisfy the Toda lattice equationṡ…”

Section: Introductionmentioning

confidence: 99%

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Ismail¹,

Koelink²,

Román³

2018

SIGMA

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Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its q-analogue. The resulting expansion formulas are made explicit for several families corresponding to measures with infinite support, including the Wilson and Askey-Wilson polynomials. An integrated version gives the possibility to give alternate expression for orthogonal polynomials with respect to a modified weight. This gives expansions for polynomials, such as Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big q-Jacobi polynomials and big q-Laguerre polynomials. We show that one can find expansions for the orthogonal polynomials corresponding to the Toda-modification of the weight for the classical polynomials that correspond to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre, Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials.

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Section: Introductionmentioning

confidence: 99%

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Ismail¹,

Koelink²,

Román³

2018

SIGMA

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A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

Meulens¹

2022

AIP Advances

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Repetitive curling of the incompressible viscid Navier–Stokes differential equation leads to a higher-order diffusion equation. Substituting this equation into the Navier–Stokes differential equation transposes the latter into the Korteweg–De Vries–Burgers-equation with the Weierstrass p-function as the soliton solution. However, a higher-order derivative of the studied variable produces the so-called N-soliton solution, which is comparable with the N-soliton solution of the Kadomtsev–Petviashvili equation. Experiments have made it clear that the system behaves like a coupled (an)harmonic oscillator on a discrete collapsed-state level. The streamlines obtained are derivatives of the error function as a function of the obtained Lax functional of the particle filaments dynamics induced by the (hypothetical) Calogero–Moser many-body system with elliptical potential and are the so-called Hermite functions. Hermite tried to introduce doubly periodic Hermite functions (the so-called Hermite problem) using coefficients related to the Weierstrass p-function. A solution-sensitive analysis of the incompressible viscid Navier–Stokes equation is performed using the Lamb vector. Cases with a meaningful potential-energy contribution require a particle interaction model with an N-soliton solution using a hierarchy-like solution of the Kadomtsev–Petviashvili equation. A three-soliton solution is emulated for the cylinder-wake problem. Our analytical results are put in perspective by comparison with two well-studied benchmark cases of fluid dynamics: the cylinder-wake problem and the driven-lid problem. The time-average velocity distribution (limit of streamline patterns) is consistent with published results and is enclosed in an asymmetrical lemniscate.

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Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

Dominici

2020

Bull. Math. Sci.

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Elliptic solutions of the Toda chain and a generalization of the Stieltjes–Carlitz polynomials

Cited by 3 publications

References 35 publications

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions

A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation

Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

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