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Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation
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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.
Title: Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation
Description:
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.
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