Quantum and electrostatic models of Chebyshev polynomials of type IV

Abstract In this work, we study the Chebyshev polynomials of the fourth kind as solutions of a singular Sturm–Liouville problem and establish their differential, orthogonality, and extremal properties. We derive a Rodrigues-type formula and a generating function, thereby establishing a direct connection with an extended Poisson kernel. Furthermore, we formulate an electrostatic model, in the spirit of Stieltjes’ classical interpretation, in which the zeros of these polynomials are characterized as the equilibrium positions of identical unit charges on (−1, 1) interacting through a logarithmic potential and subject to fixed external charges at the endpoints. We prove that the unique configuration minimizing the total energy coincides with the set of zeros of these polynomials. In addition, we introduce a principal SU (2)-bundle over R 3 ${\mathbb{R}}^{3}$ to model internal spin degrees of freedom. Within this geometric framework, the polynomials arise naturally as components of eigenfunctions of the squared spin operator for spin- 1 2 $\frac{1}{2}$ particles in the internal fiber. These results highlight the interplay between quantum spin geometry, orthogonal polynomials, and harmonic analysis on principal bundles.