<i>q</i>-special functions with |<i>q</i>| = 1 and their application to discrete integrable systems

Nishizawa, Michitomo

doi:10.1088/0305-4470/34/48/327

Cited by 2 publications

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Section: Introductionmentioning

confidence: 99%

“…In the case of discrete soliton equations, such vacuum solutions and soliton solutions constructed from the vacuum have not yet been studied in depth. So far only a few examples of solutions of nonautonomous partial difference equations are known [4,5].The purpose of this article is to construct a discrete analog of the N-soliton solution for the Toda molecule equation. We take an appropriate vacuum and show how the vacuum solution works to satisfy the boundary condition and which type of special function appears to express the soliton solution.…”

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Some Aspects of the Toda Molecule

2005

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Abstract.A q-discrete analog of the Toda molecule equation and its N-soliton solution are constructed by using the bilinear method. The solution is expressed in the Casorati determinant form whose elements are given in terms of the q-orthogonal polynomials.2000 Mathematics Subject Classification. 35Q58, 33C47. Introduction.Recently the discrete integrable systems have attracted lots of interest since their deep relations with various physical problems and numerical schemes have been found. For example, the discrete Toda equation is nothing but the scheme of qd-algorithm for computing poles of meromorphic functions [1, 2]. One of the interesting features of a Toda equation is that it allows several types of determinant expressions of solutions according to the boundary conditions. The infinite Toda lattice has the soliton solutions which are expressed by Wronski or Gram determinant, while in the finite or semi-infinite lattice case, i.e., the so-called Toda molecule case, the general solution is given in terms of a Wronski determinant. In 1998, Nakamura [3] showed that even in the molecule case, the soliton solutions do exist and he obtained their Gram and Casorati determinant expressions whose elements are given by the Gauss hypergeometric functions. The nontrivial vacuum solution plays the essential role for the soliton solutions to satisfy the boundary condition. It is known that the nontrivial vacuum is also crucial for the similarity reduction to nonautonomous systems. In the case of discrete soliton equations, such vacuum solutions and soliton solutions constructed from the vacuum have not yet been studied in depth. So far only a few examples of solutions of nonautonomous partial difference equations are known [4,5].The purpose of this article is to construct a discrete analog of the N-soliton solution for the Toda molecule equation. We take an appropriate vacuum and show how the vacuum solution works to satisfy the boundary condition and which type of special function appears to express the soliton solution. There are many possibilities for the choice of vacuum solution. In this paper we consider the q-discrete case by choosing a trigonometric vacuum and give the Casorati determinant expression of

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Section: Introductionmentioning

confidence: 99%

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confidence: 99%

Some Aspects of the Toda Molecule

2005

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“…The hyperbolic gamma function is the important building block for q-analysis with |q| = 1. It was used in [13] and [14] to construct for |q| = 1 explicit integral solutions of the q-Bessel difference equation and of the q-hypergeometric difference equation. Ruijsenaars' [19] used hyperbolic gamma functions to construct an eigenfunction for the Askey-Wilson second order difference operator for |q| = 1 as an explicit Barnes' type integral.…”

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Askey-Wilson Functions and Quantum Groups

Stokman

Theory and Applications of Special Functions

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Eigenfunctions of the Askey-Wilson second order q-difference operator for 0 < q < 1 and |q| = 1 are constructed as formal matrix coefficients of the principal series representation of the quantized universal enveloping algebra U q (sl(2, C)). The eigenfunctions are in integral form and may be viewed as analogues of Euler's integral representation for Gauss' hypergeometric series. We show that for 0 < q < 1 the resulting eigenfunction can be rewritten as a very-well-poised 8 φ 7 -series, and reduces for special parameter values to a natural elliptic analogue of the cosine kernel. Dedicated to Mizan RahmanContents.§1 Introduction. §2 Generalized gamma functions. §3 The Askey-Wilson second order difference operator and quantum groups. §4 The Askey-Wilson function for 0 < q < 1. §5 The expansion formula and the elliptic cosine kernel. §6 The Askey-Wilson function for |q| = 1. §7 Appendix.

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q-special functions with |q| = 1 and their application to discrete integrable systems

Cited by 2 publications

References 17 publications

Some Aspects of the Toda Molecule

Some Aspects of the Toda Molecule

Askey-Wilson Functions and Quantum Groups

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