We find a new class of the Fuchsian equations, which have an algebraic geometric solutions with the parameter belonging to a hyperelliptic curve. Methods of calculating the algebraic genus of the curve, and its branching points, are suggested. Numerous examples are given. Introduction Integrability of the Heun equation with half-odd characteristic exponents dy dz + P (z) dy dz +Q(z)y = 0, (0.1)...

Time and band limiting operators are expressed as functions of the confluent Heun operator arising in spheroidal wave equation. Explicit formulas obtained when bandwidth parameter is either small or large results on complete Fourier transform recovered.

By employing a pseudoorthonormal coordinate-free approach, the Dirac equation for particles in Kerr–Newman spacetime is separated into its radial and angular parts. In massless case to which special attention given, general Heun-type equations turn their confluent form. We show how one recovers some results previously obtained literature, by other means.

The scheme developed by Hartle for describing slowly rotating bodies in 1967 was applied to the simple model of constant density by Chandrasekhar and Miller in 1974. The pivotal equation one has to solve turns out to be one of Heun’s equations. After a brief discussion of this equation and the chances of finding a closed form solution, a quickly converging series solution of it is presented. A ...

Abstract. The first order nonlinear ODE φ̇(t) + sinφ(t) = q(t), q(t) = B +A cosωt, where A, B, ω are real constants, is considered, the transformation converting it to a second order linear homogeneous ODE with polynoimial coefficients is found. The latter is identified as a particular case of the double confluent Heun equation. The series of algebraic constraints on the constant parameters is f...

We study the competitive action of magnetic field, Coulomb repulsion and space curvature on the motion of a charged particle. The three types of interaction are characterized by three basic lengths: lB the magnetic length, l0 the Bohr radius and R the radius of the sphere. The energy spectrum of the particle is found by solving a Schrödinger equation of the Heun type, using the technique of con...

The Heine–Stieltjes Theorem describes the polynomial solutions, (v, f) such that T (f) = vf , to specific second order differential operators, T , with polynomial coefficients. We extend the theorem to concern all (nondegenerate) differential operators preserving the property of having only real zeros, thus solving a conjecture of B. Shapiro. The new methods developed are used to describe intri...

The study of polynomial solutions to the classical Lamé equation in its algebraic form, or equivalently, of double-periodic solutions of its Weierstrass form has a long history. Such solutions appear at integer values of the spectral parameter and their respective eigenvalues serve as the ends of bands in the boundary value problem for the corresponding Schrödinger equation with finite gap pote...