*(English)*Zbl 0685.33009

[For the entire collection see Zbl 0638.00018.]

Stieltjes gave a characterization of the x-zeros of the Jacobi polynomial ${P}_{n}^{(\alpha ,\beta )}\left(x\right)$ 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 $(\alpha +1)/2$ and $(\beta +1)/2$ 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 $(a+1)/2$ at the origin and (c- a)/2 at the point ${a}^{-1}$ $(a>0)$ 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 $(c+1)/2$. 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 ${}_{2}{F}_{0}(-n,c+n,x)\xb7$ They go on to consider the situation of m positive point charges of strength q placed at $r{\omega}_{k}$, where ${\omega}_{k}$, $k=1,\xb7\xb7\xb7,m-1$, are the mth roots of unity and a non-negative charge of strength p at the origin. They show that n $(>m)$ 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.

##### MSC:

33C45 | Orthogonal polynomials and functions of hypergeometric type |