A machine-learning approach to weight approximation for a new family of orthogonal polynomials

Varun Kumar; K. Laxminarayanamma; Abhishek Kumar Singh; Brajesh Shukla; Saiful Rahman Mondal; Varun Kumar; K. Laxminarayanamma; Abhishek Kumar Singh; Brajesh Shukla; Saiful Rahman Mondal

doi:10.3934/math.2025843

AIMS Mathematics

2025, Volume 10, Issue 8: 18861-18886. doi: 10.3934/math.2025843

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A machine-learning approach to weight approximation for a new family of orthogonal polynomials

1.
Department of Pure and Applied Mathematics, Alliance School of Sciences, Alliance University, Bangalore 562106, Karnataka, India
2.
CSE Department, Institute of Aeronautical Engineering, Dundigal Hyderabad-500043, Telangana, India
3.
Department of Mathematics, Patna Science College, Patna University, Patna 800005, Bihar, India
4.
Department of Computer Science Engineering, SRM University Delhi–NCR, Sonepat, 131029, Haryana, India
5.
Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa 31982, Saudi Arabia

Received: 16 May 2025 Revised: 28 July 2025 Accepted: 08 August 2025 Published: 20 August 2025
MSC : 33C45, 34A30, 65H10, 68T20

This research introduces a novel two-parameter family of orthogonal polynomials that emerge as solutions to a doubly confluent Heun-type differential equation. We investigate these polynomials, examining their geometric properties and analyzing the behavior and distribution of their zeros under varying parameter conditions. Leveraging machine learning techniques, we successfully derive symbolic expressions for the corresponding weight functions associated with these orthogonal polynomials. Our numerical results demonstrate the efficacy of this approach, achieving a maximum absolute error of order $ 10^{-4} $ in weight function approximation. Furthermore, we present a comparison between our proposed model and conventional approximation methods, including cubic spline interpolation and Lagrange polynomial interpolation, highlighting the advantages of our methodology.
- Heun equation,
- recurrence relation,
- orthogonal polynomials,
- zeros,
- machine learning,
- symbolic regression,
- weight approximation
Citation: Varun Kumar, K. Laxminarayanamma, Abhishek Kumar Singh, Brajesh Shukla, Saiful Rahman Mondal. A machine-learning approach to weight approximation for a new family of orthogonal polynomials[J]. AIMS Mathematics, 2025, 10(8): 18861-18886. doi: 10.3934/math.2025843

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Abstract

This research introduces a novel two-parameter family of orthogonal polynomials that emerge as solutions to a doubly confluent Heun-type differential equation. We investigate these polynomials, examining their geometric properties and analyzing the behavior and distribution of their zeros under varying parameter conditions. Leveraging machine learning techniques, we successfully derive symbolic expressions for the corresponding weight functions associated with these orthogonal polynomials. Our numerical results demonstrate the efficacy of this approach, achieving a maximum absolute error of order $ 10^{-4} $ in weight function approximation. Furthermore, we present a comparison between our proposed model and conventional approximation methods, including cubic spline interpolation and Lagrange polynomial interpolation, highlighting the advantages of our methodology.

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