Electrostatic models for zeros of Laguerre-Sobolev polynomials

Let {$\{S_n{\}}_{n⩾0}$} be the sequence of orthogonal polynomials with respect to the Laguerre-Sobolev inner product $$⟨f,g{⟩}_S=\text{\hspace{0.17em}⁣}{\int }_0^{+\mathrm{\infty }}\text{\hspace{0.17em}⁣}f(x)g(x)x^{\alpha }{e}^{-x}dx+\sum _{j=1}^N\sum _{k=0}^{d_j}{\lambda }_{j,k}{f}^{(k)}(c_j)g^{(k)}(c_j),$$ where ${\lambda }_{j,k}⩾0$, $\alpha >-1$ and $c_i\in (-\mathrm{\infty },0)$ for $i=1,2,\dots ,N$. We provide a formula that relates the Laguerre-Sobolev polynomials $S_n$ to the standard Laguerre orthogonal polynomials. We find the ladder operators for the polynomial sequence $\{S_n{\}}_{n⩾0}$ and a second-order differential equation with polynomial coefficients for $\{S_n{\}}_{n⩾0}$. We establish a sufficient condition for an electrostatic model of the zeros of orthogonal Laguerre-Sobolev polynomials. Some examples are given where this condition is either satisfied or not.