Electrostatic Models for Zeros of Laguerre-Sobolev Polynomials

Let {Sn}n⩾0 be the sequence of orthogonal polynomials with respect to the Laguerre–Sobolev inner product

⟨f, g⟩S = 0 ∫ +ÝE; f(x)g(x)xB1;e −xdx + j=1 ∑ N k=0 ∑ dj λj,kf(k)(cj)g(k)(cj),
where λj,k ⩾ 0, α > −1 and ci ∈ (−∞, 0) for i = 1, 2, ... , N. We provide a formula that relates the Laguerre–Sobolev polynomials Sn to the classical Laguerre polynomials. We find the ladder operators for the polynomial sequence {Sn}n⩾0 and a second-order differential equation with polynomial coefficients for {Sn}n⩾0. We establish a sufficient condition for an electrostatic model of the zeros of orthogonal Laguerre–Sobolev polynomials. Some examples are given where this condition is either satisfied or not.

laguerre polynomials; sobolev orthogonality; second-order differential equation; electrostatic model

Electrostatic Models for Zeros of Laguerre-Sobolev Polynomials Articles