Affiliation:
1. Norfolk State University
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.
Subject
Geology,Ocean Engineering,Water Science and Technology
Reference21 articles.
1. A. M. Legendre, Recherches sur L’attraction des Spheroides Homogenes, Universitatsbibliothek Johann Christian Senckenberg 1785 (1785) 411–434.
2. G. Szegö, Orthogonal Polynomials, 4th Edition, American Mathematical Society, Rhode Island, 1975.
3. 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.
4. 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.
5. 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.