相关论文: Heun functions versus elliptic functions

In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M.…

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.

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 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 give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's…

In the paper we deal with the Heun functions --- solutions of the Heun equation, which is the most general Fuchsian equation of second order with four regular singular points. Despite the increasing interest to the equation and numerous…

A particular solution to the equations of motion of the Abelian Higgs model is given. The solution involves the Jacobi elliptic functions as well as the Heun functions.

A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical…

The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-Laplacian. The eigenvalue and the corresponding eigenfunction are represented in terms of common parameters, and a complete description…

We give a new derivation and characterisation of the generalised elliptic genus of Krichever-H\"ohn by means of a functional equation.

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstras P-function using two…

In the present article we introduce and study a novel type of solutions of the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Felix Klein symmetric form…

We review several results on the finite-gap potential and Heun's differential equation, and we discuss relationships among the finite-gap potential, the WKB analysis and Heun's differential equation.

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…

In the paper we consider the Heun functions, which are solutions of the equation introduced by Karl Heun in 1889. The Heun functions generalize many known special functions and appear in many fields of modern physics. Evaluation of the…

General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the "classical" special functions. In particular, an elliptic analogue of the Gauss 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…

We introduce new hypergeometric series expansions of the solutions to the general Heun equation. The form of the Gauss hypergeometric functions used as expansion function differs from that used before. We derive three such expansions and…

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 a recent paper, the canonical forms of a new multi-parameter class of Abel differential equations, so-called AIR, all of whose members can be mapped into Riccati equations, were shown to be related to the differential equations for the…