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

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.