AIMSVSW 2018: Orthogonal Polynomials pp 137-154

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

• Merlin Mouafo Wouodjié
Conference paper
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.

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