Related papers: Solutions for the General, Confluent and Biconflue…
We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations --- the generalization of the confluent hypergeometric…
We study the solutions of the wave equation where a massless scalar field is coupled to the Wahlquist metric, a type-D solution. We first take the full metric and then write simplifications of the metric by taking some of the constants in…
We find transformations of variables which preserve the form of the equation for the kernels of integral relations among solutions of the Heun equation. These transformations lead to new kernels for the Heun equation, given by single…
After a brief introduction to Heun type functions we note that the actual solutions of the eigenvalue equation emerging in the calculation of the one loop contribution to QCD from the Belavin-Polyakov-Schwarz-Tyupkin instanton and the…
A new bi-parametric $su(1,1)$ algebraization of the Heun class of equations is explored. This yields additional quasi-polynomial solutions of the form $\{z^{\alpha}P_N(z): \ \alpha \in \mathbb{C}, \ N \in \mathbb{N}_0\}$ to the General Heun…
We derive a concise closed-form solution for a linear three-term recurrence relation. Such recurrence relations are very common in the quantitative sciences, and describe finite difference schemes, solutions to problems in Markov processes…
Firstly, we construct kernels of integral relations among solutions of the confluent Heun equation (CHE) and its limit, the reduced CHE (RCHE). In both cases we generate additional kernels by systematically applying substitutions of…
We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the Gauss hypergeometric functions to…
The Heun functions satisfy linear ordinary differential equations of second order with certain singularities in the complex plane. The first order derivatives of the Heun functions satisfy linear second order differential equations with one…
By exploiting a recently developed connection between Heun's differential equation and the generalized associated Lam\'e equation, we not only recover the well known periodic solutions, but also obtain a large class of new, quasi-periodic…
The Leaver solutions in series of Coulomb wave functions for the confluent Heun equation (CHE) are given by two-sided infinite series, that is, by series where the summation index $n$ runs from minus to plus infinity [E. W. Leaver, J. Math.…
Starting from the equation obeyed by the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the Appell generalized hypergeometric functions of two variables of the fist kind. Several cases…
The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be…
In the paper, the well-known quantum mechanical problem of a spin 1/2 particle in external Coulomb potential, reduced to a system of two first-order differential equations, is studied from the point of view of possible applications of the…
Applying the approach based on the equation for the derivative, we construct several expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions. Several expansions in terms of the Appell generalized…
The main purpose of this paper is to lay the foundations of a general theory which encompasses the features of the classical Hough transform and extend them to general algebraic objects such as affine schemes. The main motivation comes from…
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…
We obtain several degenerations of the $q$-Heun equation by considering the linear $q$-difference equations associated to several $q$-Painlev\'e equations. We establish definitions of the confluent $q$-Heun equation, the biconfluent…
Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third,…
We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy…