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

Abstract

AbstractWe 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.KeywordsHypergeometric functionsOperatorsTransformationsChange of variablesExp-productGauge transformationSingularitiesGeneralized exponentsExponent differencesRational functionsZeroesPolesMathematics Subject Classification (2000)34-XX33C1033C234B3034Lxx