An electrostatic model for zeros of perturbed Laguerre polynomials

Abstract:

In this paper we consider the sequences of polynomials $\{Q_{n}^{(\alpha )}\}_{n\geq 0}$, orthogonal with respect to the inner product \[ \langle f,g\rangle _{\nu }=\int _{0}^{+\infty }f(x)g(x)d\mu (x)+\sum _{j=1}^{m}a_{j} f(c_{j})g(c_{j}), \] where $d\mu (x)=x^{\alpha }e^{-x}$ is the Laguerre measure on $\mathbb {R}_{+}$, $\alpha \! >\!-1$, $c_{j}\!<0$, $a_{j}\!>0$ and $f, g$ are polynomials with real coefficients. We first focus our attention on the representation of these polynomials in terms of the standard Laguerre polynomials. Next we find the explicit formula for their outer relative asymptotics, as well as the holonomic equation that such polynomials satisfy. Finally, an electrostatic interpretation of their zeros in terms of a logarithmic potential is presented.