Jacobi–Sobolev-type orthogonal polynomials: holonomic equation and electrostatic interpretation – a non-diagonal case

Abstract

Consider the Sobolev-type inner product where p and q are polynomials with real coefficients, α, β>−1, ℙ(x)=(p(x), p′(x)) t , and is a positive semidefinite matrix, with M 0, M 1≥0, and λ∈ℝ. We obtain an expression for the family of polynomials , orthogonal with respect to the above inner product, a connection formula that relates with some family of Jacobi polynomials and the holonomic equation that they satisfy, as well as an electrostatic interpretation of their zeros.