Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation
Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation
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Date
2023-08-06
Authors
Juan Toribio-Milane
Javier Quintero-Roba
HΓ©ctor Pijeira-Cabrera
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Abstract
We study the sequence of monic polynomials {ππ}πβ©Ύ0, ortogonal with respect to the Jacobi-Sobolev inner product β¨ π, π β©π = β«_{-1}^{1} π(π₯)π(π₯)πππΌ,π½(π₯) + β_{π=1}^{π} β_{π=0}^{ππ} ππππ,π π(π)(ππ)π(π)(ππ), where π, ππ β β€^+, ππ,π β©Ύ 0, πππΌ,π½(π₯) = (1βπ₯)^πΌ(1+π₯)^π½ππ₯, πΌ,π½ > β1, and ππβπ
β(β1,1). A connection formula that relates the Sobolev polynomials ππ with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence {ππ}β©Ύ0 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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Pijeira-Cabrera, H., Quintero-Roba, J., & Toribio-Milane, J. (2023). Differential properties of Jacobi-Sobolev polynomials and electrostatic interpretation. Mathematics, 11(3420).