SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS

A. M. Ishkhanyan

doi:10.11948/2019.118

2019 Volume 9 Issue 1

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A. M. Ishkhanyan. SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 118-139. doi: 10.11948/2019.118

Citation:

A. M. Ishkhanyan. SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS[J]. Journal of Applied Analysis & Computation, 2019, 9(1): 118-139. doi: 10.11948/2019.118

SERIES SOLUTIONS OF CONFLUENT HEUN EQUATIONS IN TERMS OF INCOMPLETE GAMMA-FUNCTIONS

A. M. Ishkhanyan^1,2,

1.
Russian-Armenian University, 0051 Yerevan, Armenia
2.
Institute for Physical Research, NAS of Armenia, 0203 Ashtarak, Armenia

Author Bio: Email address: aishkhanyan@gmail.com (A.M. Ishkhanyan)
Fund Project: This work has been supported by the Armenian State Committee of Science (SCS Grants No. 18RF-139 and 18T-1C276), the Armenian National Science and Education Fund (ANSEF Grant No. PS-4986), and the RussianArmenian (Slavonic) University at the expense of the Ministry of Education and Science of the Russian Federation

Abstract

Abstract

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 indefinite integral representation. The approach employs an auxiliary equation involving only the derivatives of a solution of the equation under consideration. Using power-series expansions of the solutions of this auxiliary equation, we construct several expansions of the four confluent Heun equations' solutions in terms of the incomplete Gamma-functions. In the cases of single- and double-confluent Heun equations the coefficients of the expansions obey four-term recurrence relations, while for the bi- and tri-confluent Heun equations the recurrence relations in general involve five terms. Other expansions for which the expansion coefficients obey recurrence relations involving more terms are also possible. The particular cases when these relations reduce to ones involving less number of terms are identified. The conditions for deriving closed-form finite-sum solutions via right-hand side termination of the constructed series are discussed.
- Confluent Heun equations /
- special functions /
- series expansions /
- multi-term recurrence relations
MSC: 33E30, 34B30, 30Bxx

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References

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