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Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation
(2023-08-06) Juan Toribio-Milane; Javier Quintero-Roba; Héctor Pijeira-Cabrera
We study the sequence of monic polynomials {𝑆𝑛}𝑛⩾0, ortogonal with respect to the Jacobi-Sobolev inner product ⟨ 𝑓, 𝑔 ⟩𝑠 = ∫_{-1}^{1} 𝑓(𝑥)𝑔(𝑥)𝑑𝜇𝛼,𝛽(𝑥) + ∑_{𝑗=1}^{𝑁} ∑_{𝑘=0}^{𝑑𝑗} 𝑑𝑗𝜆𝑗,𝑘 𝑓(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}⩾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.