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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 (α,β) (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 (α+1)/2 and (β+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, we get a “generalized dipole” at 0, the point charges becoming + and - 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)· They go on to consider the situation of m positive point charges of strength q placed at rω k , where ω 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.

Reviewer: M.E.Muldoon
MSC:
33C45Orthogonal polynomials and functions of hypergeometric type