The Heun functions have wide application in modern physics and are expected to succeed the hypergeometrical functions in the physical problems of the 21st century. The numerical work with those functions, however, is complicated and requires filling the gaps in the theory of the Heun functions and also, creating new algorithms able to work with them efficiently. We propose a new algorithm for s...

We find quasinormal modes of near extremal black branes by solving a singular boundary value problem for the Heun equation. The corresponding eigenvalues determine the dispersion law for the collective excitations in the dual strongly coupled N = 4 SYM at finite temperature. In memory of Aleksandr Sergeevich Vshivtsev

The well-known Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in the linear approximation. From a mathematical point of view it presents a particular case of the confluent Heun equation and can be solved exactly, due to recent mathematical developments. We present the basic properties of its general solution. A novel analytical approach and numerical techniques...

In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some simple examples of transformations between equations that are not Fuchsian and that generalise the Heun-toHypergeometric transformations.

Infinitely many cases for which two independent fundamental solutions of the biconfluent Heun equation can each be presented as an irreducible linear combination confluent generalized hypergeometric functions are identified. The involved functions, in general do not reduce to polynomials, such that numerator parameter (except one) exceeds a corresponding denominator by unity.