Jacobi–Sobolev Orthogonal Polynomials, Differential Properties and Structural Formulas

In this paper, we extend some differential and structural results for monic Jacobi–Sobolev orthogonal polynomials, associated with a general discrete Sobolev inner product, with a Jacobi continuous part. We consider finitely many exterior mass points and a positive semidefinite Sobolev product matrix. Using Christoffel–Darboux kernels, we derive several structure and connection formulas involving two consecutive Jacobi and Jacobi–Sobolev polynomials. This representation leads to lowering and raising operators with rational coefficients; a second-order ordinary differential equation and a three-term recurrence relation, with polynomial coefficients. These coefficients depend on n. These results extend several classical structural properties of Jacobi polynomials to a general discrete Sobolev setting.