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On the Solutions of Holonomic Third-Order Linear Irreducible Differential Equations in Terms of Hypergeometric Functions

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Book cover Orthogonal Polynomials (AIMSVSW 2018)

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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

$$\displaystyle y=~ \exp \left (\int r \,dx \right )\left (r_0S(f(x))+r_1(S(f(x)))^{\prime }+r_2(S(f(x)))^{\prime \prime }\right ). $$

An implementation of this algorithm in Maple is available.

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Notes

  1. 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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Authors and Affiliations

  1. Institute of Mathematics, University of Kassel, Kassel, Germany

    Merlin Mouafo Wouodjié

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  1. Merlin Mouafo Wouodjié
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Correspondence to Merlin Mouafo Wouodjié .

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Editors and Affiliations

  1. University of Yaoundé I, Yaoundé, Cameroon; African Institute for Mathematical Sciences, Limbe, Cameroon

    Mama Foupouagnigni

  2. Institute for Mathematics, University of Kassel, Kassel, Germany

    Wolfram Koepf

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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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