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More on electrostatic models for zeros of orthogonal polynomials. (English) Zbl 0981.42015

This is a very interesting, well written paper. The author extends results given in [F. A. Grünbaum, J. Comput. Appl. Math. 99, No. 1-2, 189-194 (1998; Zbl 0934.33012)] and [M. E. H. Ismail, Pac. J. Math. 193, 355-369 (2000; Zbl 1011.33011)].

His main results are:

1. The derivation of a second order differential equation (under mild conditions) for the polynomials ${p}_{n}\left(x\right)$, orthogonal with respect to a measure $d\mu \left(x\right)=w\left(x\right)dx+d{\mu }_{s}\left(x\right)$, where $w\left(x\right)$ is a weight function on $\left[a,b\right]$ and ${\mu }_{s}\left(x\right)={\sum }_{k\ge 0}\mu \left({\xi }_{k}\right)\epsilon \left({\xi }_{k},x\right)$ with ${\xi }_{k}\notin \left(a,b\right)$ and $\epsilon \left(u,x\right)$ an atomic measure centered at $x=u$.

2. Uniqueness of the equilibrium position of $n$ movable charges in $\left(a,b\right)$ in the presence of the usual external field connected with the measure (built from $v\left(x\right)$ given by $w\left(x\right)=exp\left(-v\left(x\right)\right)$ and another logarithmic term), which is attained at the zeros of ${p}_{n}\left(x\right)$, provided that the particle interaction law obeys a logarithmic potential.

Moreover, the equilibrium energy is calculated in terms of the external potential and weight function, taken at the zeros of ${p}_{n}\left(x\right)$ and the coefficients from the three term recurrence relation (via the discriminant of the orthogonal polynomials).

Finally, explicit results are given in the case of the Koornwinder polynomials (Jacobi measure on $\left(-1,1\right)$, with Dirac measures added in $x=±1$) and in the case of Ginsburg-Landau (${v}^{\text{'}}\left(x\right)=4x\left({x}^{2}-c\right)$) and generalized Jacobi potentials ($w\left(x\right)=c{x}^{\alpha }{\prod }_{j=0}^{m}{\left({c}_{j}-x\right)}^{{\beta }_{j}}$).

##### MSC:
 42C05 General theory of orthogonal functions and polynomials 33C45 Orthogonal polynomials and functions of hypergeometric type