Electrostatic Models for Zeros of Laguerre–Sobolev Polynomials

Let \(\{S_n\}_{n\geqslant 0}\) ${\{S_n\}}_{n⩾0}$ be the sequence of orthogonal polynomials with respect to the Laguerre–Sobolev inner product \(\begin{aligned} \langle f,g\rangle _S =\!\int _{0}^{+\infty }\! f(x) g(x)x^{\alpha }e^{-x}dx+\sum _{j=1}^{N}\sum _{k=0}^{d_j}\lambda _{j,k} f^{(k)}(c_j)g^{(k)}(c_j), \end{aligned}\) $$\begin{array}{c}\hfill {⟨f,g⟩}_S={\int }_0^{+\infty }f(x)g(x)x^{\alpha }{e}^{-x}dx+\sum _{j=1}^N\sum _{k=0}^{d_j}{\lambda }_{j,k}{f}^{(k)}(c_j)g^{(k)}(c_j),\end{array}$$ where \(\lambda _{j,k}\geqslant 0\) ${\lambda }_{j,k}⩾0$ , \(\alpha >-1\) $\alpha >-1$ and \(c_i \in (-\infty , 0)\) $c_i\in (-\infty ,0)$ for \(i=1,2,\dots ,N\) $i=1,2,\cdots ,N$ . We provide a formula that relates the Laguerre–Sobolev polynomials \(S_n\) $S_n$ to the classical Laguerre polynomials. We find the ladder operators for the polynomial sequence \(\{S_n\}_{n\geqslant 0}\) ${\{S_n\}}_{n⩾0}$ and a second-order differential equation with polynomial coefficients for \(\{S_n\}_{n\geqslant 0}\) ${\{S_n\}}_{n⩾0}$ . 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.