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

摘要

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 is an element of {F-0(2), F-1(2), F-2(2), B-nu(2)} where F-p(q) with p is an element of{0, 1, 2}, q = 2, is the generalized hypergeometric function, and B-nu(2)(x) = (B-nu(root x)(2) with B-nu Bessel function (see (Abramowitz and Stegun, 1972)). That means we find rational functions r, r(0), r(1), r(2), fsuch that the solution of L will be of the formy = exp(integral rdx) (r(0)S(f(x)) + r(1)(S(f(x)))' + r(2)(S(f(x)))'').An implementation of those algorithms in Maple is available. (C) 2019 Elsevier Ltd. All rights reserved.