Recurrence coefficients of generalized Meixner polynomials and Painlevé equations
TL;DR: In this paper, a semi-classical version of the Meixner weight depending on two parameters and the associated set of orthogonal polynomials was considered, and it was shown that the coefficients appearing in this relation satisfy a discrete Painleve equation, which is a limiting case of an asymmetric dPIV equation.
Abstract: We consider a semi-classical version of the Meixner weight depending on two parameters and the associated set of orthogonal polynomials. These polynomials satisfy a three-term recurrence relation. We show that the coefficients appearing in this relation satisfy a discrete Painleve equation, which is a limiting case of an asymmetric dPIV equation. Moreover, when viewed as functions of one of the parameters, they satisfy one of Chazy's second-degree Painleve equations, which can be reduced to the fifth Painleve equation PV.
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28 Dec 2017TL;DR: In this article, a survey of recent results and an outline of their close relationship with orthogonal polynomials are presented. But the authors do not discuss the relationship between orthogonality and Painleve equations.
Abstract: There are a number of intriguing connections between Painleve equations and orthogonal polynomials, and this book is one of the first to provide an introduction to these. Researchers in integrable systems and non-linear equations will find the many explicit examples where Painleve equations appear in mathematical analysis very useful. Those interested in the asymptotic behavior of orthogonal polynomials will also find the description of Painleve transcendants and their use for local analysis near certain critical points helpful to their work. Rational solutions and special function solutions of Painleve equations are worked out in detail, with a survey of recent results and an outline of their close relationship with orthogonal polynomials. Exercises throughout the book help the reader to get to grips with the material. The author is a leading authority on orthogonal polynomials, giving this work a unique perspective on Painleve equations.
83 citations
TL;DR: The authors showed that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painleve equation when viewed as functions of one of the parameters in the weight.
Abstract: We show that the coefficients of the three-term recurrence relation for orthogonal polynomials with respect to a semi-classical extension of the Laguerre weight satisfy the fourth Painleve equation when viewed as functions of one of the parameters in the weight. We compare different approaches to derive this result, namely, the ladder operators approach, the isomonodromy deformations approach and combining the Toda system for the recurrence coefficients with a discrete equation. We also discuss a relation between the recurrence coefficients for the Freud weight and the semi-classical Laguerre weight and show how it arises from the Backlund transformation of the fourth Painleve equation.
60 citations
Cites background from "Recurrence coefficients of generali..."
...• PIV and the weights |x−t|ρe−x 2 in [8], xαe−x 2+tx, x > 0, and |x|2α+1e−x4+tx2 (in this paper); • the discrete Meixner weight (γ)kc/(k!(β)k), c, β, γ > 0, and PV [3, 15]; • PV and the weights (1−ξθ(x−t))|x−t|αxμe−x, where θ is the Heaviside function in [17], xα(1−x)βe−t/x in [7], (1+x)α(1−x)βe−tx, x ∈ (−1, 1), in [1]; • PVI and the weights xα(1 − x)β(A + Bθ(x − t)), x ∈ [0, 1], in [11], (1− x)αxβ(t− x)γ , x ∈ [−1, 1], in [26]; see also [18] for more examples and applications in random matrix theory....
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...The recurrence coefficients a(2)n and bn can always be written [12] as ratios of Hankel determinants containing the moments of the orthogonality measure (see also in [3])....
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TL;DR: In this paper, the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painleve equation is discussed.
Abstract: We discuss the relationship between the recurrence coefficients of orthogonal polynomials with respect to a semiclassical Laguerre weight and classical solutions of the fourth Painleve equation. We show that the coefficients in these recurrence relations can be expressed in terms of Wronskians of parabolic cylinder functions that arise in the description of special function solutions of the fourth Painleve equation.
57 citations
TL;DR: In this paper, it was shown that the spectrum of the Rabi model coincides with the support of the discrete Stieltjes integral measure in the orthogonality relations of recently introduced orthogonal polynomials.
40 citations
TL;DR: In this article, it was shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials.
Abstract: We investigate semi-classical generalizations of the Charlier and Meixner polynomials, which are discrete orthogonal polynomials that satisfy three-term recurrence relations. It is shown that the coefficients in these recurrence relations can be expressed in terms of Wronskians of modified Bessel functions and confluent hypergeometric functions, respectively for the generalized Charlier and generalized Meixner polynomials. These Wronskians arise in the description of special function solutions of the third and fifth Painleve equations.
29 citations
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21 Nov 2005
TL;DR: In this paper, the authors present q-series Preliminaries, q-Summation theorems, Exponential and q-bessel functions, and Askey-Wilson polynomials.
Abstract: Foreword Preface 1. Preliminaries 2. Orthogonal polynomials 3. Differential equations, Discriminants and electrostatics 4. Jacobi polynomials 5. Some inverse problems 6. Discrete orthogonal polynomials 7. Zeros and inequalities 8. Polynomials orthogonal on the unit circle 9. Linearization, connections and integral representations 10. The Sheffer classification 11. q-series Preliminaries 12. q-Summation theorems 13. Some q-Orthogonal polynomials 14. Exponential and q-bessel functions 15. The Askey-Wilson polynomials 16. The Askey-Wilson operators 17. q-Hermite polynomials on the unit circle 18. Discrete q-orthogonal polynomials 19. Fractional and q-fractional calculus 20. Polynomial solutions to functional equations 21. Some indeterminate moment problems 22. The Riemann-Hilbert problem for orthogonal polynomials 23. Multiple orthogonal polynomials 24. Research problems Bibliography Index Author index.
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"Recurrence coefficients of generali..." refers background in this paper
...Painlevé and Gambier proved that up to Möbius transformations, only 50 equations of the form (9) exist which satisfy the Painlevé property [14, 32, 33]....
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