Abstract
We present here an algorithm that combines change of variables, exp-product and gauge transformation to represent solutions of a given irreducible third-order linear differential operator L, with rational function coefficients and without Liouvillian solutions, in terms of functions \(S\in \left \{{{ }_1F_1}^2, ~{{ }_0F_2}, ~_1F_2, ~_2F_2\right \}\) where pF q with p ∈{0, 1, 2}, q ∈{1, 2}, is the generalized hypergeometric function. That means we find rational functions r, r 0, r 1, r 2, f such that the solution of L will be of the form
An implementation of this algorithm in Maple is available.
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Notes
- 1.
To generate this differential equation, we use the hsum package from Wolfram Koepf (see [4]).
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Acknowledgements
This work has been supported by a DAAD scholarship (German Academic Exchange Service) and the University of Kassel by a “Promotions-Abschlussstipendium”. All these institutions receive my sincere thanks.
Many thanks to the organizers of the AIMS–Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications, Douala, October 5–12, 2018.
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Mouafo Wouodjié, M. (2020). On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_8
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