zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $P\sb n\sp{(\alpha,\beta)}(x)$ 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.
Reviewer: M.E.Muldoon

33C45Orthogonal polynomials and functions of hypergeometric type