Maxwell orthogonal polynomials
Yıl 2024,
, 93 - 113, 16.12.2024
Angel Alvarez-paredes
,
Ruymán Cruz-barroso
,
Francisco Marcellán
Öz
In the framework of the theory of semiclassical linear functionals in this contribution we deal with the sequence of orthogonal polynomials associated with the linear functional $ \langle{L, p}\rangle = \int_{0} ^{\infty} p(x) e^{- x^2}dx,$ where $p\in \mathbb{P},$ the linear space of polynomials with complex coefficients. The class of $L$ is one and we deduce a differential/difference equation (structure relation) for the sequence of orthogonal polynomials. The Laguerre-Freud equations that the coefficients of the three term recurrence relation satisfy are deduced. The connection with discrete Painlev\'e IV equations is emphasized. Finally, we analyze the lowering and raising operators (ladder operators) for such polynomials in order to find a second order linear differential equation they satisfy. As a consequence, an electrostatic interpretation of their zeros is formulated.
Destekleyen Kurum
Ministerio de Ciencia, Innovación y Universidades of Spain
Proje Numarası
PID2021- 122154NB-I00
Kaynakça
- S. Belmehdi: On semi-classical linear functionals of class s = 1. Classification and integral representations, Indag. Math. (New Series), 3 (3) (1992), 253–275.
- S. Belmehdi: A. Ronveaux, Laguerre-Freud’s equations for the recurrent coefficients of semi-classical orthogonal polynomials, J. Approx. Theory, 76 (3) (1994), 351–368.
- Y. Chen, M. E. H. Ismail: Ladder operators and differential equations for orthogonal polynomials, J. Phys. A: Math. Gen., 30 (22) (1997), 7817–7829.
- Y. Chen, G. Pruessner: Orthogonal polynomials with discontinuous weights, J. Phys. A: Math. Gen., 38 (12) (2005), L191–L199.
- T. S. Chihara: An Introduction to Orthogonal Polynomials, New York, Dover Publications (2011).
- A. S. Clarke, B. Shizgal: On the Generation of Orthogonal Polynomials Using Asymptotic Methods for Recurrence Coefficients, J. Comput. Phys., 104 (1) (1993), 140–149.
- D. Dominici, F. Marcellán: Truncated Hermite polynomials, J. Difference Equ. Appl., 29 (7) (2023), 701–732.
- J. C. García-Ardila, F. Marcellán and M. E. Marriaga: Orthogonal Polynomials and Linear Functionals. An Algebraic Approach and Applications, Berlin, European Mathematical Society (2021).
- M. Landreman, D. R. Ernst: New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions, J. Comput. Phys., 243 (2013), 130–150.
- S. Lyu, Y. Chen: Gaussian unitary ensembles with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems, Stud. Appl. Math., 146 (1) (2021), 118–138.
- A. P. Magnus: Freud’s equations for orthogonal polynomials as discrete Painlevé equations, in Symmetries and Integrability of Difference Equations, P. A. Clarkson and F.W. Nijhoff, Eds., Cambridge University Press., London Math. Soc. Lect. Note Ser., vol 255, 228–343, 1999.
- P. Maroni: Prolégomènes à l’ étude des polynômes orthogonaux semi-classiques, Ann. Mat. Pura Appl. (4), 149 (1987), 165–184.
- P. Maroni: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, in Orthogonal Polynomials and Their Applications, C. Brezinski et al., Eds., IMACS Ann. Comput. Appl. Math., vol. 9, 95–130, Baltzer, Basel (1991).
- T. Sánchez-Vizuet, A. J. Cerfon: Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion, Plasma Phys. Control. Fusion, 60 (2) (2018), Article ID: 025018.
- B. Shizgal: A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems, J. Comput. Phys., 41 (2) (1981), 309–328.
- J. Shohat: A differential equation for orthogonal polynomials, Duke Math. J., 5 (1939), 401–417.
- W. Van Assche: Orthogonal Polynomials and Painlevé Equations, Cambridge, Cambridge University Press (2018).
- W. Van Assche: Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations, in Orthogonal Polynomials and Their Applications, Mama Foupouagnigni and Wolfram Koepf, Eds., 629–683, Cham Springer International Publishing (2020).
- M. Vanlessen: Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory, Constr. Approx., 25 (2) (2007), 125–175.
Yıl 2024,
, 93 - 113, 16.12.2024
Angel Alvarez-paredes
,
Ruymán Cruz-barroso
,
Francisco Marcellán
Proje Numarası
PID2021- 122154NB-I00
Kaynakça
- S. Belmehdi: On semi-classical linear functionals of class s = 1. Classification and integral representations, Indag. Math. (New Series), 3 (3) (1992), 253–275.
- S. Belmehdi: A. Ronveaux, Laguerre-Freud’s equations for the recurrent coefficients of semi-classical orthogonal polynomials, J. Approx. Theory, 76 (3) (1994), 351–368.
- Y. Chen, M. E. H. Ismail: Ladder operators and differential equations for orthogonal polynomials, J. Phys. A: Math. Gen., 30 (22) (1997), 7817–7829.
- Y. Chen, G. Pruessner: Orthogonal polynomials with discontinuous weights, J. Phys. A: Math. Gen., 38 (12) (2005), L191–L199.
- T. S. Chihara: An Introduction to Orthogonal Polynomials, New York, Dover Publications (2011).
- A. S. Clarke, B. Shizgal: On the Generation of Orthogonal Polynomials Using Asymptotic Methods for Recurrence Coefficients, J. Comput. Phys., 104 (1) (1993), 140–149.
- D. Dominici, F. Marcellán: Truncated Hermite polynomials, J. Difference Equ. Appl., 29 (7) (2023), 701–732.
- J. C. García-Ardila, F. Marcellán and M. E. Marriaga: Orthogonal Polynomials and Linear Functionals. An Algebraic Approach and Applications, Berlin, European Mathematical Society (2021).
- M. Landreman, D. R. Ernst: New velocity-space discretization for continuum kinetic calculations and Fokker–Planck collisions, J. Comput. Phys., 243 (2013), 130–150.
- S. Lyu, Y. Chen: Gaussian unitary ensembles with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems, Stud. Appl. Math., 146 (1) (2021), 118–138.
- A. P. Magnus: Freud’s equations for orthogonal polynomials as discrete Painlevé equations, in Symmetries and Integrability of Difference Equations, P. A. Clarkson and F.W. Nijhoff, Eds., Cambridge University Press., London Math. Soc. Lect. Note Ser., vol 255, 228–343, 1999.
- P. Maroni: Prolégomènes à l’ étude des polynômes orthogonaux semi-classiques, Ann. Mat. Pura Appl. (4), 149 (1987), 165–184.
- P. Maroni: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, in Orthogonal Polynomials and Their Applications, C. Brezinski et al., Eds., IMACS Ann. Comput. Appl. Math., vol. 9, 95–130, Baltzer, Basel (1991).
- T. Sánchez-Vizuet, A. J. Cerfon: Pseudo spectral collocation with Maxwell polynomials for kinetic equations with energy diffusion, Plasma Phys. Control. Fusion, 60 (2) (2018), Article ID: 025018.
- B. Shizgal: A Gaussian quadrature procedure for use in the solution of the Boltzmann equation and related problems, J. Comput. Phys., 41 (2) (1981), 309–328.
- J. Shohat: A differential equation for orthogonal polynomials, Duke Math. J., 5 (1939), 401–417.
- W. Van Assche: Orthogonal Polynomials and Painlevé Equations, Cambridge, Cambridge University Press (2018).
- W. Van Assche: Orthogonal and Multiple Orthogonal Polynomials, Random Matrices, and Painlevé Equations, in Orthogonal Polynomials and Their Applications, Mama Foupouagnigni and Wolfram Koepf, Eds., 629–683, Cham Springer International Publishing (2020).
- M. Vanlessen: Strong Asymptotics of Laguerre-Type Orthogonal Polynomials and Applications in Random Matrix Theory, Constr. Approx., 25 (2) (2007), 125–175.