A class of Truncated Freud polynomials

2025/10/10 by García-Ardila, Juan Carlos, Marcellán, Francisco, Marriaga, Misael E.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2510.09214

Abstract

Consider the following truncated Freud linear functional u_z depending on a parameter z, \langleu_z,p⟩=∫₀^∞ p(x)e^-zx4dx, zgt;0. The aim of this work is to analyze the properties of the sequence of orthogonal polynomials (P_n)_{n≥ 0} with respect to u_z. Such a linear functional is semiclassical and, as a consequence, we get the system of nonlinear difference equations (Laguerre-Freud equations) that the coefficients of the three-term recurrence satisfy. The asymptotic behavior of such coefficients is given. On the other hand, the raising and lowering operators associated with such a linear functional are obtained, and thus a second-order linear differential equation of holonomic type that (P_n)_{n≥ 0} satisfies is deduced. From this fact, an electrostatic interpretation of their zeros is given. Finally, some illustrative numerical tests concerning the behavior of the least and greatest zeros of these polynomials are presented.

Citations

• Symmetrization process and truncated orthogonal polynomials
• Truncated Hermite polynomials
• Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems
• Orthogonal polynomials with discontinuous weights
• Ladder operators and differential equations for orthogonal polynomials
• Painlev 'e-type differential equations for the recurrence coefficients\n of semi-classical orthogonal polynomials.
• A differential equation for orthogonal polynomials

Related

• Symmetric Truncated Freud polynomials
• Symmetrization process and truncated orthogonal polynomials
• Laguerre-Freud equations for Generalized Hahn polynomials of type I
• Truncated Hermite polynomials
• Asymptotics of Polynomials Orthogonal With Respect to a Generalized Freud Weight With Application to Special Function Solutions of Painlevé-IV