Differential Properties of Jacobi-Sobolev Polynomials and Electrostatic Interpretation

We study the sequence of monic polynomials { S n } n ⩾ 0 , orthogonal with respect to the Jacobi-Sobolev inner product ⟨ f , g ⟩ s = ∫ − 1 1 f ( x ) g ( x ) d μ α , β ( x ) + ∑ j = 1 N ∑ k = 0 d j λ j , k f ( k ) ( c j ) g ( k ) ( c j ) , where N , d j ∈ Z + , λ j , k ⩾ 0 , d μ α , β ( x ) = ( 1 − x ) α ( 1 + x ) β d x , α , β > − 1 , and c j ∈ R ∖ ( − 1 , 1 ) . A connection formula that relates the Sobolev polynomials S n with the Jacobi polynomials is provided, as well as the ladder differential operators for the sequence { S n } 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.