On the solutions of holonomic third-order linear irreducible differential equations in terms of hypergeometric functions

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∈{F20,F21,F22,Bˇν2} where Fqp with p∈{0,1,2}, q=2, is the generalized hypergeometric function, and Bˇν2(x)=(Bν(x))2 with Bν a Bessel function (see (Abramowitz and Stegun, 1972)). That means we find rational functions r,r0,r1,r2,f such that the solution of L will be of the formy=exp⁡(∫rdx)(r0S(f(x))+r1(S(f(x)))′+r2(S(f(x)))″).An implementation of those algorithms in Maple is available.