On the solutions of holonomic third-order linear irreducible differential equations in terms of hypergeometric functions. (November 2020) - Mouafo Wouodjié, Merlin; Koepf, Wolfram
Abstract: In this paper we present algorithms that combine 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 ∈ { F 2 0, F 2 1, F 2 2, B ˇ ν 2 } where F q p with p ∈ { 0, 1, 2 }, q = 2, is the generalized hypergeometric function, and B ˇ ν 2 ( x ) = ( B ν ( x ) ) 2 with B ν a Bessel function (see (Abramowitz and Stegun, 1972 )). 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   y = exp ⁡ ( ∫ r d x ) ( r 0 S ( f ( x ) ) + r 1 ( S ( f ( x ) ) ) ′ + r 2 ( S ( f ( x ) ) ) ″ ) . An implementation of those algorithms in Maple is available.