Painlevé, Heun and the Symmetric Nonsingular Jacobi Polynomials

In this paper, our aim is to investigate orthogonal polynomials and Hankel determinants that are generated by a symmetric nonsingular Jacobi weight. The methods utilized, such as ladder operators and Coulomb fluid, significantly contribute to a more profound comprehension of the properties of the ensemble and their relationships with well-established mathematical frameworks. By adapting the ladder operators to the monic orthogonal polynomials concerning this weight and carefully monitoring the evolution of parameters in the orthogonality relation, we discover a significant connection between one of the auxiliary quantities {f₍} (t) and the Painlevé V equation, following a suitable transformation of variables. Through the utilization of the Coulomb fluid, we derive the large n asymptotic expansion of the recurrence coefficient, aiding in the reduction of the second-order differential equation satisfied by the monic orthogonal polynomials associated with this weight to the analogous general Heun equation. Furthermore, we identify a novel quantity linked to the logarithmic derivative of the Hankel determinant that satisfies both a differential equation and a difference equation. These analyses allow us to establish connections among diverse mathematical entities and offer a more profound insight into the underlying mathematical structures at play.