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Electrostatic interpretation of zeros. (English) Zbl 0685.33009
Orthogonal polynomials and their applications, Proc. Int. Symp., Segovia/Spain 1986, Lect. Notes Math. 1329, 241-250 (1988).

[For the entire collection see Zbl 0638.00018.]

Stieltjes gave a characterization of the x-zeros of the Jacobi polynomial ${P}_{n}^{\left(\alpha ,\beta \right)}\left(x\right)$ as the positions of equilibrium of n unit electrical charges in the interval (-1,1) in the field (with logarithmic potential) generated by positive charges $\left(\alpha +1\right)/2$ and $\left(\beta +1\right)/2$ at 1 and -1. The present authors extend this idea to polynomials with complex zeros and electrical charges in the complex plane. First they consider point charges of strength $\left(a+1\right)/2$ at the origin and (c- a)/2 at the point ${a}^{-1}$ $\left(a>0\right)$ on the real axis, where c is not zero or a negative integer. As $a\to \infty$, we get a “generalized dipole” at 0, the point charges becoming $+\infty$ and -$\infty$ respectively while their sum has the constant value $\left(c+1\right)/2$. The authors show that if n positive unit charges are in equilibrium in the two-dimensional field due to this dipole, they must coincide with the (mostly complex) zeros of the Bessel polynomial ${}_{2}{F}_{0}\left(-n,c+n,x\right)·$ They go on to consider the situation of m positive point charges of strength q placed at $r{\omega }_{k}$, where ${\omega }_{k}$, $k=1,···,m-1$, are the mth roots of unity and a non-negative charge of strength p at the origin. They show that n $\left(>m\right)$ positive free unit charges will be in equilibrium in the resultant field if and only if they coincide with the zeros of a polynomial solution of a certain differential equation. Several special cases and confluences of this result are then considered.

Reviewer: M.E.Muldoon
##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type
##### Keywords:
electrostatic interpretation; Bessel polynomial