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.
Categories: math.CA math.CV