Related papers: Quantum Field Theory Applications of Heun Type Fun…
We examine the power-series solutions and the series solutions in terms of the Hermite functions for the biconfluent Heun equation. Infinitely many cases for which a solution of the biconfluent equation is presented as an irreducible linear…
In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points --- two of them are regular and the third one is irregular of rank 1. The purpose of the work is to…
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…
In this paper, a quantum dot mathematical model based on a two-dimensional Schr\"odinger equation assuming the 1/r inter-electronic potential is revisited. Generally, it is argued that the solutions of this model obtained by solving a…
The energy eigenvalues of a Dirac particle for the hyperbolic-type potential field have been computed approximately. It is obtained a transcendental function of energy, $\mathcal{F}(E)$, by writing in terms of confluent Heun functions. The…
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…
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…
Sch\"afke and Schmidt established that the asymptotics of the coefficients of the local solution to some linear differential equation is related to global structures of solutions. The Heun class equations have the accessory parameters, and…
We obtain special solutions of the $q$-Heun equation which are expressed as finite summations of $q$-hypergeometric functions. These solutions are obtained by considering the $q$-integral transformations of the polynomial-type solutions.
The effects on the non-relativistic dynamics of a system compound by two electrons interacting by a Coulomb potential and with an external harmonic oscillator potential, confined to move in a two dimensional Euclidean space, are…
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…
We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schr\"odinger equation with an inverted quasi-exactly…
The local Heun solution is the unique solution to Heun's equation which is analytic in the unit disk centered at $0\in\mathbb{C}$ and taking the value $1$ at the center of the disk. In this paper, as an application of the theory of…
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…
We find confluent Heun solutions to the radial equations of two Halilsoy-Badawi metrics. For the first metric, we studied the radial part of the massless Dirac equation and for the second case, we studied the radial part of the massless…
The $q$-Heun equation is a $q$-difference analogue of Heun's differential equation. We review several solutions of Heun's differential equation and investigate polynomial-type solutions of $q$-Heun equation. The limit $q\to 1$ corresponding…
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 consider the non-stationary Heun equation, also known as quantum Painlev\'e VI, which has appeared in different works on quantum integrable models and conformal field theory. We use a generalized kernel function identity to transform the…
It is shown that the regular-at-infinity solution of the 1D Schrodinger equation with the hyperbolic Poschl-Teller (h-PT) potential with integer parameters is expressible in terms of a n-order Heun polynomial in y=thr at an arbitrary…
Heun differential equations are the most general second order Fuchsian equations with four regular singularities. An explicit integral series representation of Heun functions involving only elementary integrands has hitherto been unknown…