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 JacobiSobolev inner product < f,g > s = integral(1)(-1) f (x)g(x) d mu(alpha,beta)(x) + Sigma (N)(dj)(j=1) lambda(j,k),f(k) (c(j))g((k))(cj), where N, d(j) is an element of Z(+), lambda(j,k) >= 0, d mu(alpha,beta)(x) = (1-x)(alpha)(1 + x)beta (dx), alpha, beta > -1, and c(j) is an element of R backslash(-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 {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.

Differential properties of Jacobi-Sobolev polynomials and electrostatic interpretation Articles