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Exceptional orthogonal polynomials and the Darboux transformation [PDF]
Journal of Physics A: Mathematical and Theoretical, 2010 We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation. We also show that the shape-invariance of these new polynomial families is a direct consequence of the ...
Gómez-Ullate Otaiza, David +2 more
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Darboux transforms on Band Matrices, Weights and associated Polynomials
, 2000 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 ...
Adler, Mark, van Moerbeke, Pierre
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Exceptional Legendre Polynomials and Confluent Darboux Transformations [PDF]
Symmetry, Integrability and Geometry: Methods and Applications, 2021 Exceptional orthogonal polynomials are families of orthogonal polynomials that arise as solutions of Sturm-Liouville eigenvalue problems. They generalize the classical families of Hermite, Laguerre, and Jacobi polynomials by allowing for polynomial sequences that miss a finite number of ''exceptional'' degrees.
Garcia Ferrero, María Ángeles +2 more
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Seeking Darboux Polynomials [PDF]
Acta Applicandae Mathematicae, 2014 We introduce several techniques which allow to simplify the expression of the cofactor of Darboux polynomials of polynomial differential systems in R^n. We apply these techniques to some well-known systems when n = 2, 3, 4. We also propose a general method for computing Darboux polynomials in the plane.
Ferragut, Antoni +1 more
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Liouvillian Integrability Versus Darboux Polynomials [PDF]
Qualitative Theory of Dynamical Systems, 2016 Agraïments: FEDER/UNAB10-4E-378. The second author is supported by Portuguese National Funds through FCT - Fundacâo para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD. The third author is partially supported by NNSF of China grant number 11271252, by RFDP of Higher Education of China grant number 20110073110054, and by ...
Llibre, Jaume +2 more
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Darboux Integrability of a Generalized 3D Chaotic Sprott ET9 System
مجلة بغداد للعلوم, 2022 In this paper, the first integrals of Darboux type of the generalized Sprott ET9 chaotic system will be studied. This study showed that the system has no polynomial, rational, analytic and Darboux first integrals for any value of .
Adnan Ali Jalal +2 more
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Dual optimal design and the Christoffel–Darboux polynomial [PDF]
Optimization Letters, 2021 The purpose of this short note is to show that the Christoffel-Darboux polynomial, useful in approximation theory and data science, arises naturally when deriving the dual to the problem of semi-algebraic D-optimal experimental design in statistics. It uses only elementary notions of convex analysis.
de Castro, Yohann +3 more
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Acta Polytechnica, 2022 The paper advances Odake and Sasaki’s idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the
Gregory Natanson
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On generalized Heun equation with some mathematical properties
Acta Polytechnica, 2022 We study the analytic solutions of the generalized Heun equation, (α0 + α1 r + α2 r2 + α3 r3) y′′ + (β0 + β1 r + β2 r2) y′ + (ε0 + ε1 r) y = 0, where |α3| + |β2|≠ 0, and {αi}3i=0, {βi}2i=0, {εi}1i=0 are real parameters.
Nasser Saad
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Darboux Polynomials for Lotka–Volterra Systems in Three Dimensions [PDF]
Journal of Nonlinear Mathematical Physics, 2021 We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the parameters, and give the explicit form of the corresponding cofactors.
Christodoulides, Y. T. +3 more
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