Electrostatic models for zeros of Laguerre-Sobolev polynomials

2023/08/11 by Díaz-González, Abel, Pijeira-Cabrera, Héctor, Quintero-Roba, Javier
#30C15(Primary) 42C0 #33C45 #33C47 #82B23 (Secondary) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2308.06304

Abstract

Let \S_n\_{n\geqslant 0} be the sequence of orthogonal polynomials with respect to the Laguerre-Sobolev inner product ⟨ f,g⟩_S = ∫₀^+∞ f(x) g(x)x^αe^-xdx+∑_j=1^N∑_k=0^d_jλ_j,k f^(k)(c_j)g^(k)(c_j), where λ_j,k\geqslant 0, α>-1 and c_i ∈ (-∞, 0) for i=1,2,…,N. We provide a formula that relates the Laguerre-Sobolev polynomials S_n to the standard Laguerre orthogonal polynomials. We find the ladder operators for the polynomial sequence \S_n\_{n\geqslant 0} and a second-order differential equation with polynomial coefficients for \S_n\_{n\geqslant 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.

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