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

  • Merlin Mouafo WouodjiéEmail author
Conference paper
  • 45 Downloads
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)

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 pFq with p ∈{0, 1, 2}, q ∈{1, 2}, is the generalized hypergeometric function. That means we find rational functions r, r0, r1, r2, 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.

Keywords

Hypergeometric functions Operators Transformations Change of variables Exp-product Gauge transformation Singularities Generalized exponents Exponent differences Rational functions Zeroes Poles 

Mathematics Subject Classification (2000)

34-XX 33C10 33C2 34B30 34Lxx 

Notes

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.

References

  1. 1.
    R. Debeerst, Solving differential equations in terms of Bessel functions. Master’s Thesis, Universität Kassel (2007)Google Scholar
  2. 2.
    R. Debeerst, M. van Hoeij, W. Koepf, Solving differential equations in terms of Bessel functions, in Proceedings of the 2008 International Symposium on Symbolic and Algebraic Computation (ISSAC’08) (2008), pp. 39–46Google Scholar
  3. 3.
    P. Horn, Faktorisierung in Schief-Polynomringen. Ph.D. Thesis, Universität Kassel (2008). https://kobra.bibliothek.uni-kassel.de/handle/urn:nbn:de:hebis:34-2009030226513
  4. 4.
    W. Koepf, Hypergeometric Summation—An Algorithmic Approach to Summation and Special Function Identities (Springer, Berlin, 2014)CrossRefGoogle Scholar
  5. 5.
    J. Kovacic, An algorithm for solving second-order linear homogeneous equations. J. Symb. Comput. 2, 3–43 (1986)MathSciNetCrossRefGoogle Scholar
  6. 6.
    M. Mouafo Wouodjié, On the solutions of holonomic third-order linear irreducible differential equations in terms of hypergeometric functions. Ph.D. Thesis, Universität Kassel (2018). https://kobra.bibliothek.uni-kassel.de/handle/urn:nbn:de:hebis:34-2018060655613
  7. 7.
    M. Mouafo Wouodjié, W. Koepf, On the solutions of holonomic third-order linear irreducible differential equations in terms of hypergeometric functions. J. Symb. Comput. (2019). https://doi.org/10.1016/j.jsc.2019.08.002
  8. 8.
    M.F. Singer, Solving homogeneous linear differential equations in terms of second order linear differential equations. Am. J. Math. 107, 663–696 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    M. van der Put, M.F. Singer, Galois Theory of Linear Differential Equations. Comprehensive Studies in Mathematics, vol. 328 (Springer, Berlin, 2003)Google Scholar
  10. 10.
    M. van Hoeij, Factorization of linear differential operators. Ph.D. Thesis, Universitijt Nijmegen (1996)Google Scholar
  11. 11.
    M. van Hoeij, Rational solutions of the mixed differential equation and its application to factorization of differential operators, in Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation (ISSAC’96) (1996), pp. 219–225Google Scholar
  12. 12.
    M. van Hoeij, Factorization of linear differential operators with rational functions coefficients. J. Symb. Comput. 24, 237–561 (1997)Google Scholar
  13. 13.
    M. van Hoeij, Q. Yuan, Finding all Bessel type solutions for linear differential equations with rational function coefficients, in Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC’10) (2010), pp. 37–44Google Scholar
  14. 14.
    Q. Yuan, Finding all Bessel type solutions for linear differential equations with rational function coefficients. Ph.D. Thesis, Florida State University (2012)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of KasselKasselGermany

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