On Laguerre-Sobolev type orthogonal polynomials, zeros and electrostatic interpretation

Abstract

We study the sequence of monic polynomials orthogonal with respect to inner product $$\langle p,q\rangle = \int_0^\infty p(x)q(x)e^{-x}x^{\alpha}\;dx +Mp(\zeta )q(\zeta )+Np^\prime(\zeta )q^\prime(\zeta),$$ where $\alpha >-1$, $M\geq 0$, $N\geq 0$, $\zeta < 0$, and $p$ and $q$ are polynomials with real coefficients. We deduce some interlacing properties of their zeros and, by using standard methods, we find a second order linear differential equation satisfied by the polynomials and discuss an electrostatic model of their zeros. doi:10.1017/S1446181113000308