Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 085, 18 pages      arXiv:1207.5302      https://doi.org/10.3842/SIGMA.2012.085
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Global Solutions of Certain Second-Order Differential Equations with a High Degree of Apparent Singularity

Ryu Sasaki a and Kouichi Takemura b
a) Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
b) Department of Mathematics, Faculty of Science and Technology, Chuo University, 1-13-27 Kasuga, Bunkyo-ku Tokyo 112-8551, Japan

Received July 24, 2012, in final form November 07, 2012; Published online November 15, 2012

Abstract
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent −2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.

Key words: multi-indexed orthogonal polynomials; solvable systems; Fuchsian differential equations; Heun's equation; apparent singularities; high characteristic exponents.

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