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 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 and 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 at the origin and (c- a)/2 at the point on the real axis, where c is not zero or a negative integer. As , we get a “generalized dipole” at 0, the point charges becoming and - respectively while their sum has the constant value . 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 They go on to consider the situation of m positive point charges of strength q placed at , where , , are the mth roots of unity and a non-negative charge of strength p at the origin. They show that n 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.