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 $P\sb n\sp{(\alpha,\beta)}(x)$ 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 $a\sp{-1}$ $(a>0)$ 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 $\sb 2F\sb 0(-n,c+n,x).$ They go on to consider the situation of m positive point charges of strength q placed at $r\omega\sb k$, where $\omega\sb k$, $k=1,...,m-1$, are the mth roots of unity and a non-negative charge of strength p at the origin. They show that n $(>m)$ 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.