相关论文: New solutions of Heun general equation

Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a…

The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This…

We provide a set of diagonals of simple rational functions of three and four variables that are squares of Heun functions. These Heun functions obtained through creative telescoping, turn out to be either pullbacked $_2F_1$ hypergeometric…

We present a conspicuous number of indefinite integrals involving Heun functions and their products obtained by means of the Lagrangian formulation of a general homogeneous linear ordinary differential equation. As a by-product we also…

The Heun's equation with its four regular singularities emerges in many applications in science. Despite the growing interest of the scientific community, the literature has many gaps in conceptual mathematical aspects of this equation.…

Just as with the Gauss hypergeometric function, particular cases of the local Heun function can be Liouvillian (that is, "elementary") functions. One way to obtain these functions is by pull-back transformations of Gauss hypergeometric…

In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…

We give examples of where the Heun function exists as solutions of wave equations encountered in general relativity. While the Dirac equation written in the background of Nutku helicoid metric yields Mathieu functions as its solutions in…

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in…

The Heun's equation is the Fuchsian equation of second order with four regular singularities. Heun functions generalize well-known special functions such as Spheroidal Wave, Lam\'{e}, Mathieu, hypergeometric-type functions, etc. The…

We apply solutions of Heun's general equation to the stationary Schr\"odinger equation with two quasi-exactly solvable elliptic potentials which depend on a real parameter $\ell$. We get finite-series solutions from power series expansions…

We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the…

An approach to construction of static models is demonstrated for a fluid ball. Five examples are considered. Two of them are exact solutions of the Einstein equations; the other three are connected with the Airy special functions, the…

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…

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes…

Integral relations and transformation rules are used to obtain, out of an asymptotic solution, a new group of four pairs of solutions to the double-confluent Heun equation. Each pair presents the same series coefficients but has solutions…

A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the…

In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that…

When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…

We derive five classes of quantum time-dependent two-state models solvable in terms of the double confluent Heun functions, five other classes solvable in terms of the biconfluent Heun functions, and a class solvable in terms of the…