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28 March 2024
 
  » arxiv » math.CA/0206285

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Algebraic Solutions of the Lamé Equation, Revisited
Robert S. Maier ;
Date 27 Jun 2002
Journal J. Differential Equations 198 (2004) 16-34. DOI: 10.1016/j.jde.2003.06.006
Subject Classical Analysis and ODEs; Mathematical Physics MSC-class: 34A20 (Primary) 33E10,14H05 (Secondary) | math.CA math-ph math.MP
AffiliationUniversity of Arizona
AbstractA minor error in the necessary conditions for the algebraic form of the Lamé equation to have a finite projective monodromy group, and hence for it to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On algebraic solutions of Lamé’s differential equation", J. Differential Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6 from an integer; this possibility was missed. The omission affects a recent result on the monodromy of the Weierstrass form of the Lamé equation. [See R. C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric, Riemann, and Lamé equations", J. Symbolic Computation 28 (1999), 521-545.] The Weierstrass form, which is a differential equation on an elliptic curve, may have, after all, an octahedral projective monodromy group.
Source arXiv, math.CA/0206285
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