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} ;$$⟨f,g{⟩}_{\mathsf{s}}={\int }_{-1}^{1}f(x)g(x)d{\mu }^{\alpha ,\beta }(x)+\sum _{j=1}^N\sum _{k=0}^{d_j}{\lambda }_{j,k}{f}^{(k)}(c_j)g^{(k)}(c_j),$$ ; where $N,d_j\in {\text{ZZ}}_{+}$, ${\lambda }_{j,k}⩾0$, $d{\mu }^{\alpha ,\beta }(x)=(1-x{)}^{\alpha }(1+x{)}^{\beta }dx$, $\alpha ,\beta >-1$, and $c_j\in \text{RR}\setminus (-1,1)$. 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 $\{S_n{\}}_{n⩾0}$ 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.