Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions

@article{Ishkhanyan2014ExpansionsOT,
  title={Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions},
  author={T. A. Ishkhanyan and Artur M. Ishkhanyan},
  journal={arXiv: Classical Analysis and ODEs},
  year={2014}
}
We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions. The coefficients of the expansions in general obey three-term recurrence relations defining double-sided infinite series; however, four-term and two-term relations are also possible in particular cases. The conditions for left- and/or right-side termination of the derived series are discussed. 
Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions
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
Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations
We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions
Hypergeometric expansions of the solutions of the general Heun equation governed by two-term recurrence relations for expansion coefficients
We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions the forms of which differ from
Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation
Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are
Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients
We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from
Appell Hypergeometric Expansions of the Solutions of the General Heun Equation
Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun
SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS
  • A. Ishkhanyan
  • Mathematics
    Journal of Applied Analysis & Computation
  • 2019
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
Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions
Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun
...
...

References

SHOWING 1-10 OF 22 REFERENCES
Incomplete beta-function expansions of the solutions to the confluent Heun equation
Several expansions of the solutions to the confluent Heun equation in terms of incomplete beta functions are constructed. A new type of expansion involving certain combinations of the incomplete beta
Appell Hypergeometric Expansions of the Solutions of the General Heun Equation
Starting from a second-order Fuchsian differential equation having five regular singular points, an equation obeyed by a function proportional to the first derivative of the solution of the Heun
New solutions of Heun general equation
We show that in four particular cases the derivative of the solution of Heun general equation can be expressed in terms of a solution to another Heun equation. Starting from this property, we use the
On Certain Solutions for Confluent and Double-Confluent Heun Equations
This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric
THE INTEGRABILITY OF THE TWO-STATE PROBLEM IN TERMS OF CONFLUENT HYPERGEOMETRIC FUNCTIONS
The reduction of the two-state problem to the confluent hypergeometric equation via complex transformations of independent variables is considered. It is shown that all the known analytically
New solutions to the confluent Heun equation and quasiexact solvability
We construct new solutions in series of confluent hypergeometric functions for the confluent Heun equation (CHE). Some of these solutions are applied to the one-dimensional stationary Schr\"{o}dinger
Confluent Heun equations: convergence of solutions in series of coulomb wavefunctions
The Leaver solutions in series of Coulomb wavefunctions for the confluent Heun equation are given by two-sided infinite series, that is, by series where the summation index n runs from minus to plus
On the global representation of the solutions of second-order linear differential equations having an irregular singularity of rank one in :3WKby series in terms of confluent hypergeometric functions
A general second-order linear differential equation having an irregular singularity of rank one in $\infty $ is considered. It is shown that the solutions of this equation can be represented by
Potentials of the Heun class
We review different methods of generating potentials such that the one-dimensional Schrodinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with
...
...