Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

Abstract

We study the sequence of monic polynomials {Sn}n⩾0, orthogonal with respect to the Jacobi-Sobolev inner product ⟨f,g⟩s=∫−11f(x)g(x)dμα,β(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where N,dj∈Z+, λj,k⩾0, dμα,β(x)=(1−x)α(1+x)βdx, α,β>−1, and cj∈R∖(−1,1). A connection formula that relates the Sobolev polynomials Sn with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {Sn}n⩾0 and a second-order differential equation with a polynomial coefficient that they satisfied. We give sufficient conditions under which the zeros of a wide class of Jacobi-Sobolev polynomials can be interpreted as the solution of an electrostatic equilibrium problem of n unit charges moving in the presence of a logarithmic potential. Several examples are presented to illustrate this interpretation.

Asymptotic for Orthogonal Polynomials with Respect to a Rational Modification of a Measure Supported on the Semi-Axis

“…4) (or ( [8] Th. 3) for (−∞, +∞)) proves a quite general version of the relative asymptotic formula (4). In this case, if the modification function, ρ, is a nonnegative function on [0, +∞) in L 1 (µ), such that there exists an algebraic polynomial G and k ∈ N for which |G|ρ/(1…”

“…These asymptotic results are of interest for the electrostatic interpretation of zeros of Jacobi-Sobolev polynomials (cf. [4]).…”