Differential properties of Jacobi-Sobolev polynomials and electrostatic interpretation

We study the sequence of monic polynomials $\{S_n\}_{n\geqslant 0}$, orthogonal with respect to the Jacobi-Sobolev inner {product} \;$$\langle f,g\rangle_{\mathsf{s}}= \int_{-1}^{1} f(x) g(x)\, d\mu^{\alpha,\beta}(x)+\sum_{j=1}^{N}\sum_{k=0}^{d_j}\lambda_{j,k} f^{(k)}(c_j)g^{(k)}(c_j),$$ \; where $N,d_j \in \ZZ_+$, $\lambda_{j,k}\geqslant 0$, $d\mu^{\alpha,\beta}(x)=(1-x)^{\alpha}(1+x)^{\beta} dx$, $\alpha,\beta>-1$, and $c_j\in\RR\setminus (-1,1)$. 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 $\{S_n\}_{n\geqslant 0}$ 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.