In this paper we will explore the 2D system describing trapped ionic in quadrapole field with a superposition of rationally symmetric hexapole and octopole fields for meromorphic integrability. We use Lyapunov's Ziglin-Morales-Ruiz-Ramis's classical methods proofs. The monodromy group normal variational equation (NVE), which is Heun equation, studied commutativity. Some modern results from theo...

We give a list of Heun equations which are Picard-Fuchs associated to families of algebraic varieties. Our list is based on the classification of families of elliptic curves with four singular fibers done by Herfurtner. We also show that pullbacks of hypergeometric functions by rational Belyi functions with restricted ramification data give rise to Heun equations.

We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integr...

We compute the Green’s function for the Hodge Laplacian on the symmetric spaces M × Σ, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and Σ is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distrib...

An expression of the special value ζQ(2) is obtained in [2] in terms of a certain contour integral using the solution of a singly confluent type Heun differential equation. It would be indicated that these special values are complicated enough and highly transcendental as reflecting the transcendence of the spectra of the non-commutative harmonic oscillator. ∗The research of the author is suppo...