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A Unified Approach to Computing the Zeros of Orthogonal Polynomials

Year 2023, Issue: 45, 30 - 45, 31.12.2023

Ridha Moussa James Tipton

https://doi.org/10.53570/jnt.1350502

Abstract

We present a unified approach to calculating the zeros of the classical orthogonal polynomials and discuss the electrostatic interpretation and its connection to the energy minimization problem. This approach works for the generalized Bessel polynomials, including the normalized reversed variant, as well as the Viet\'e--Pell and Viet\'e--Pell--Lucas polynomials. We briefly discuss the electrostatic interpretation for each aforesaid case and some recent advances. We provide zeros and error estimates for various cases of the Jacobi, Hermite, and Laguerre polynomials and offer a brief discussion of how the method was implemented symbolically and numerically with Maple. In conclusion, we provide possible avenues for future research.

Keywords

Orthogonal polynomials zeros electrostatic interpretation

References

  • A. M. Legendre, Recherches sur L’attraction des Spheroides Homogenes, Universitatsbibliothek Johann Christian Senckenberg 1785 (1785) 411–434.
  • G. Szegö, Orthogonal Polynomials, 4th Edition, American Mathematical Society, Rhode Island, 1975.
  • A. Alhaidari, Representation of the Quantum Mechanical Wavefunction by Orthogonal Polynomials in the Energy and Physical Parameters, Communications in Theoretical Physics 72 (1) (2019) 015104 15 pages.
  • T. M. Dunster, A. Gil, D. Ruiz-Antolin, J. Segura, Computation of the Reverse Generalized Bessel Polynomials and Their Zeros, Computational and Mathematical Methods 3 (6) (2021) e1198 12 pages.
  • B. Kuloğlu, E. Özkan, A. G. Shannon, Incomplete Generalized Vieta–Pell and Vieta–Pell–Lucas Polynomials, Notes on Number Theory and Discrete Mathematics 27 (4) (2021) 245–256.
  • D. Tasci, F. Yalcin, Vieta-Pell and Vieta-Pell-Lucas Polynomials, Advances in Difference Equations 2013 (2013) Article Number 224 8 pages.
  • M. E. H. Ismail, X.-S. Wang, On Quasi-Orthogonal Polynomials: Their Differential Equations, Discriminants and Electrostatics, Journal of Mathematical Analysis and Applications 474 (2) (2019) 1178–1197.
  • M. E. H. Ismail, An Electrostatics Model for Zeros of General Orthogonal Polynomials, Pacific Journal of Mathematics 193 (2) (2000) 355–369.
  • M. E. H. Ismail, More on Electrostatic Models for Zeros of Orthagonal Polynomials, Numerical Functional Analysis and Optimization 21 (1) (2007) 191–204.
  • A. N. Lowan, N. Davids, A. Levenson., Table of the Zeros of the Legendre Polynomials of Order 1-16 and the Weight Coefficients for Gauss’ Mechanical Quadrature Formula, Bulletin of the American Mathematical Society 48 (10) (1942) 739–743.
  • R. E. Greenwood, J. J. Miller, Zeros of the Hermite Polynomials and Weights for Gauss’ Mechanical Quadrature Formula, Bulletin of the American Mathematical Society 54 (1948) 765–769.
  • H. E. Salzer, R. Zucker, Table of the Zeros and Weight Factors of the First Fifteen Laguerre Polynomials, Bulletin of the American Mathematical Society 55 (10) (1949) 1004–1012.
  • H. L. Krall, O. Frink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Transactions of the American Mathematical Society 65 (1) (1949) 100–115.
  • L. Pasquini, Polynomial Solutions to Second Order Linear Homogeneous Ordinary Differential Equations. Properties and Approximation, Calcolo 26 (1989) 167–183.
  • L. Pasquini, On the Computation of the Zeros of the Bessel Polynomials, in: R. V. M. Zahar (Ed.), Approximation and Computation: A Festschrift in Honor of Walter Gautschi, Vol. 119 of ISNM International Series of Numerical Mathematics, Birkhauser, Boston, 1994, pp. 511–534.
  • L. Pasquini, Accurate Computation of the Zeros of the Generalized Bessel Polynomials, Numerische Mathematik 86 (3) (2000) 507–538.
  • S. Steinerberger, Electrostatic Interpretation of Zeros of Orthogonal Polynomials, Proceedings of the American Mathematical Society 146 (12) (2018) 5323–5331.
  • J. E. Marsden, M. J. Hoffman, Elementary Classical Analysis, 2nd Edition, W. H. Freeman, San Francisco, 1993.
  • F. Marcellan, A. Martinez-Finkelshtein, P. Martinez-Gonzalez, Electrostatic Models for Zeros of Polynomials: Old, New, and Some Open Problems, Journal of Computational and Applied Mathematics 207 (2) (2007) 258–272.
  • F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, M. A. McClain, NIST Digital Library of Mathematical Functions (2010), http://dlmf.nist.gov/, Accessed 20 Nov 2023 to Release 1.1.0 of 2020-12-15.
  • W. M. Abd-Elhameed, A. Napoli, Some Novel Formulas of Lucas Polynomials via Different Approaches, Symmetry 15 (1) (2023) 185 19 pages.

Year 2023, Issue: 45, 30 - 45, 31.12.2023

Ridha Moussa James Tipton

https://doi.org/10.53570/jnt.1350502

Abstract

References

  • A. M. Legendre, Recherches sur L’attraction des Spheroides Homogenes, Universitatsbibliothek Johann Christian Senckenberg 1785 (1785) 411–434.
  • G. Szegö, Orthogonal Polynomials, 4th Edition, American Mathematical Society, Rhode Island, 1975.
  • A. Alhaidari, Representation of the Quantum Mechanical Wavefunction by Orthogonal Polynomials in the Energy and Physical Parameters, Communications in Theoretical Physics 72 (1) (2019) 015104 15 pages.
  • T. M. Dunster, A. Gil, D. Ruiz-Antolin, J. Segura, Computation of the Reverse Generalized Bessel Polynomials and Their Zeros, Computational and Mathematical Methods 3 (6) (2021) e1198 12 pages.
  • B. Kuloğlu, E. Özkan, A. G. Shannon, Incomplete Generalized Vieta–Pell and Vieta–Pell–Lucas Polynomials, Notes on Number Theory and Discrete Mathematics 27 (4) (2021) 245–256.
  • D. Tasci, F. Yalcin, Vieta-Pell and Vieta-Pell-Lucas Polynomials, Advances in Difference Equations 2013 (2013) Article Number 224 8 pages.
  • M. E. H. Ismail, X.-S. Wang, On Quasi-Orthogonal Polynomials: Their Differential Equations, Discriminants and Electrostatics, Journal of Mathematical Analysis and Applications 474 (2) (2019) 1178–1197.
  • M. E. H. Ismail, An Electrostatics Model for Zeros of General Orthogonal Polynomials, Pacific Journal of Mathematics 193 (2) (2000) 355–369.
  • M. E. H. Ismail, More on Electrostatic Models for Zeros of Orthagonal Polynomials, Numerical Functional Analysis and Optimization 21 (1) (2007) 191–204.
  • A. N. Lowan, N. Davids, A. Levenson., Table of the Zeros of the Legendre Polynomials of Order 1-16 and the Weight Coefficients for Gauss’ Mechanical Quadrature Formula, Bulletin of the American Mathematical Society 48 (10) (1942) 739–743.
  • R. E. Greenwood, J. J. Miller, Zeros of the Hermite Polynomials and Weights for Gauss’ Mechanical Quadrature Formula, Bulletin of the American Mathematical Society 54 (1948) 765–769.
  • H. E. Salzer, R. Zucker, Table of the Zeros and Weight Factors of the First Fifteen Laguerre Polynomials, Bulletin of the American Mathematical Society 55 (10) (1949) 1004–1012.
  • H. L. Krall, O. Frink, A New Class of Orthogonal Polynomials: The Bessel Polynomials, Transactions of the American Mathematical Society 65 (1) (1949) 100–115.
  • L. Pasquini, Polynomial Solutions to Second Order Linear Homogeneous Ordinary Differential Equations. Properties and Approximation, Calcolo 26 (1989) 167–183.
  • L. Pasquini, On the Computation of the Zeros of the Bessel Polynomials, in: R. V. M. Zahar (Ed.), Approximation and Computation: A Festschrift in Honor of Walter Gautschi, Vol. 119 of ISNM International Series of Numerical Mathematics, Birkhauser, Boston, 1994, pp. 511–534.
  • L. Pasquini, Accurate Computation of the Zeros of the Generalized Bessel Polynomials, Numerische Mathematik 86 (3) (2000) 507–538.
  • S. Steinerberger, Electrostatic Interpretation of Zeros of Orthogonal Polynomials, Proceedings of the American Mathematical Society 146 (12) (2018) 5323–5331.
  • J. E. Marsden, M. J. Hoffman, Elementary Classical Analysis, 2nd Edition, W. H. Freeman, San Francisco, 1993.
  • F. Marcellan, A. Martinez-Finkelshtein, P. Martinez-Gonzalez, Electrostatic Models for Zeros of Polynomials: Old, New, and Some Open Problems, Journal of Computational and Applied Mathematics 207 (2) (2007) 258–272.
  • F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, M. A. McClain, NIST Digital Library of Mathematical Functions (2010), http://dlmf.nist.gov/, Accessed 20 Nov 2023 to Release 1.1.0 of 2020-12-15.
  • W. M. Abd-Elhameed, A. Napoli, Some Novel Formulas of Lucas Polynomials via Different Approaches, Symmetry 15 (1) (2023) 185 19 pages.
There are 21 citations in total.

Details

Primary Language English
Subjects Ordinary Differential Equations, Difference Equations and Dynamical Systems
Journal Section Research Article
Authors

Ridha Moussa This is me 0009-0009-4936-8555

James Tipton 0000-0002-4291-5984

Early Pub Date December 30, 2023
Publication Date December 31, 2023
Submission Date August 26, 2023
Published in Issue Year 2023 Issue: 45

Cite

APA Moussa, R., & Tipton, J. (2023). A Unified Approach to Computing the Zeros of Orthogonal Polynomials. Journal of New Theory(45), 30-45. https://doi.org/10.53570/jnt.1350502
AMA Moussa R, Tipton J. A Unified Approach to Computing the Zeros of Orthogonal Polynomials. JNT. December 2023;(45):30-45. doi:10.53570/jnt.1350502
Chicago Moussa, Ridha, and James Tipton. “A Unified Approach to Computing the Zeros of Orthogonal Polynomials”. Journal of New Theory, no. 45 (December 2023): 30-45. https://doi.org/10.53570/jnt.1350502.
EndNote Moussa R, Tipton J (December 1, 2023) A Unified Approach to Computing the Zeros of Orthogonal Polynomials. Journal of New Theory 45 30–45.
IEEE R. Moussa and J. Tipton, “A Unified Approach to Computing the Zeros of Orthogonal Polynomials”, JNT, no. 45, pp. 30–45, December 2023, doi: 10.53570/jnt.1350502.
ISNAD Moussa, Ridha - Tipton, James. “A Unified Approach to Computing the Zeros of Orthogonal Polynomials”. Journal of New Theory 45 (December 2023), 30-45. https://doi.org/10.53570/jnt.1350502.
JAMA Moussa R, Tipton J. A Unified Approach to Computing the Zeros of Orthogonal Polynomials. JNT. 2023;:30–45.
MLA Moussa, Ridha and James Tipton. “A Unified Approach to Computing the Zeros of Orthogonal Polynomials”. Journal of New Theory, no. 45, 2023, pp. 30-45, doi:10.53570/jnt.1350502.
Vancouver Moussa R, Tipton J. A Unified Approach to Computing the Zeros of Orthogonal Polynomials. JNT. 2023(45):30-45.
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