### Abstract

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 equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

Original language | English |
---|---|

Pages (from-to) | 445-459 |

Number of pages | 15 |

Journal | Constructive Approximation |

Volume | 49 |

Issue number | 3 |

DOIs | |

Publication status | Published - 15 Jun 2019 |

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### Keywords

- Heun equation
- Linear ordinary differential equation
- Recurrence relations
- Series expansions
- Special functions

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

### Cite this

*Constructive Approximation*,

*49*(3), 445-459. https://doi.org/10.1007/s00365-018-9424-8

**Appell Hypergeometric Expansions of the Solutions of the General Heun Equation.** / Ishkhanyan, A. M.

Research output: Contribution to journal › Article

*Constructive Approximation*, vol. 49, no. 3, pp. 445-459. https://doi.org/10.1007/s00365-018-9424-8

}

TY - JOUR

T1 - Appell Hypergeometric Expansions of the Solutions of the General Heun Equation

AU - Ishkhanyan, A. M.

PY - 2019/6/15

Y1 - 2019/6/15

N2 - 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 equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

AB - 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 equation, we construct several expansions of the solutions of the general Heun equation in terms of Appell generalized hypergeometric functions of two variables of the first kind. Several cases when the expansions reduce to those written in terms of simpler mathematical functions such as the incomplete Beta function or the Gauss hypergeometric function are identified. The conditions for deriving finite-sum solutions via termination of the series are discussed. In general, the coefficients of the expansions obey four-term recurrence relations; however, there exist certain choices of parameters for which the recurrence relations involve only two terms, though not necessarily successive. For such cases, the coefficients of the expansions are explicitly calculated and the general solution of the Heun equation is constructed in terms of the Gauss hypergeometric functions.

KW - Heun equation

KW - Linear ordinary differential equation

KW - Recurrence relations

KW - Series expansions

KW - Special functions

UR - http://www.scopus.com/inward/record.url?scp=85044764702&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044764702&partnerID=8YFLogxK

U2 - 10.1007/s00365-018-9424-8

DO - 10.1007/s00365-018-9424-8

M3 - Article

VL - 49

SP - 445

EP - 459

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 3

ER -