相关论文: Computations of general Heun functions from their …
We present a simple systematic algorithm for construction of expansions of the solutions of ordinary differential equations with rational coefficients in terms of mathematical functions having indefinite integral representation. The…
We study Heun's differential equation in the case that one of the singularities is apparent. In particular we conjecture a relationship with generalized hypergeometric differential equation and establish it in some cases. We apply our…
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…
The present article reveals important properties of the confluent Heun's functions. We derive a set of novel relations for confluent Heun's functions and their derivatives of arbitrary order. Specific new subclasses of confluent Heun's…
The cases when the equation for the derivative of the confluent Heun function has only three singularities (in general, the equation has four such points) are examined. It is shown that this occurs only in three specific cases. Further, it…
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…
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…
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…
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 construct several expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta functions and the Appell generalized hypergeometric functions of two variables of the fist kind. The coefficients of different…
We give closed forms for several families of Heun functions related to classical entropies. By comparing two expressions of the same Heun function, we get several combinatorial identities generalizing some classical ones.
The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric_2F_1,_1F_1 and_0F_1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice…
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 obtain some new formulae to compute the first derivative of confluent and biconfluent Heun functions under the minimal assumption of fixing only one parameter. These results together with the Lagrangian formulation of a general…
In this paper, we give the matrix version of Horn's hypergeometric function and its confluent cases. We also discuss the regions of convergence, the system of matrix differential equations of bilateral type, differential formulae and…
We find kernel functions of the $q$-Heun equation and its variants. We apply them to obtain $q$-integral transformations of solutions to the $q$-Heun equation and its variants. We discuss special solutions of the $q$-Heun equation from the…
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…
The confluent Heun equation is one of 4 confluent forms of Heun's differential equation in which is the Fuchsian equation of second order with four regular singularities. A confluent Heun function is applicable to diverse areas such as…
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…
By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before…