On the structure of the singularities of the Heun’s differential equations.

3 July 2014 a general meeting of AMU was held. Prof. Artur Ishkhanyan from the Institute for Physical Research, NAS of Armenia (Ashtarak) have presented a talkOn the structure of the singularities of the Heun’s differential equations.

Abstract: The general Heun equation presents a natural generalization of the Gauss hypergeometric equation. It has four regular singular points and allows four different coalescence procedures for singularities and thus leads to four different confluent Heun equations. These equations are widely faced in current physics and mathematics research and the functions defined by the solutions of these equations are believed to gradually become a part of the next generation of the standard special functions of mathematical physics.
However, the Heun equations are much less studied than their hypergeometric relatives; the analytical difficulties which occur when addressing these equations are greater by at least an order of magnitude. For instance, the solutions are no more expressed in terms of definite or contour integrals involving simpler mathematical functions. Another major point is that the power-series expansions are no more governed by familiar two-term recurrence relations between the successive coefficients of the expansion. As a result, the coefficients of the series are no more calculated explicitly and complications arise in studying of their convergence as well as in studying the connection problems for different expansions.
Analyzing the structure of the singularities of the Heun equations, we note that the equation obeyed by the derivative of a solution of these equations is a more complicated differential equation in general having an extra regular singular point. The characteristic exponents of this additional singularity are 0 and 2, and the singularity is located at a point of the complex z-plain defined solely by the accessory parameter of the Heun equations. Using the mentioned property, we construct several expansions of the solutions of the general and confluent Heun equations in terms of the Goursat generalized hypergeometric functions and the Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to ones written in terms of simpler mathematical functions such as the incomplete beta function or the hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed.

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