We use classical Jacobi polynomials to identify the equilibrium configurations of charged particles confined to the unit circle. Our main result unifies two theorems from a 1986 paper of Forrester and Rogers.
Topics
Electrostatics
REFERENCES
1.
2.
T.
Stieltjes
, “Sur certains polynômes qui vérifient une équation différentielle linéaire du second ordre et sur la theorie des fonctions de lamé
,” Acta Math.
6
(1
), 321
–326
(1885
).3.
T.
Stieltjes
, “Sur quelques théorèms d’algèbre
,” C. R. Acad. Sci., Paris
100
, 439
–440
(1885
); Oeuvres Complètes 1, 440–441.4.
T.
Stieltjes
, “Sur les polynômes de Jacobi
,” C. R. Acad. Sci., Paris
100
, 620
–622
(1885
); Oevres Complètes 1, 442–444.5.
G.
Szegő
, Orthogonal Polynomials
, 4th ed. (American Mathematical Society, Colloquium Publications
, Providence, RI
, 1975
), Vol. XXIII.6.
M. E. H.
Ismail
, “An electrostatics model for zeros of general orthogonal polynomials
,” Pac. J. Math.
193
(2
), 355
–369
(2000
).7.
F.
Marcellán
, A.
Martínez-Finkelshtein
, and P.
Martínez-González
, “Electrostatic models for zeros of polynomials: Old, new, and some open problems
,” J. Comput. Appl. Math.
207
(2
), 258
–272
(2007
).8.
P. J.
Forrester
and J. B.
Rogers
, “Electrostatics and the zeros of the classical polynomials
,” SIAM J. Math. Anal.
17
, 461
–468
(1986
).9.
A.
Grinshpan
, “A minimum energy problem and Dirichlet spaces
,” Proc. Am. Math. Soc.
130
(2
), 453
–460
(2002
).10.
A.
Grinshpan
, “Electrostatics, hyperbolic geometry and wandering vectors
,” J. London Math. Soc.
69
(1
), 169
–182
(2004
).11.
B.
Simanek
, “An electrostatic interpretation of the zeros of paraorthogonal polynomials on the unit circle
,” SIAM J. Math. Anal.
48
(3
), 2250
–2268
(2016
).12.
B.
Simon
, Convexity: An Analytic Viewpoint
, Cambridge Tracts in Mathematics Vol. 187 (Cambridge University Press
, Cambridge
, 2011
).© 2020 Author(s).
2020
Author(s)
You do not currently have access to this content.
