Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one.

*(English)*Zbl 0919.34012“A sequence of re-expansions is developed for the remainder terms in the well-known Poincaré series expansions of the solutions to homogeneous linear differential equations of higher order in the neighborhood of an irregular singularity of rank one. These re-expansions are a series whose terms are a product of Stokes multipliers, coefficients of the original Poincaré series expansions, and certain multiple integrals, the so-called hyperterminants. Each step of the process reduces the estimate of the error term by an exponentially small factor.

The method is based on the Borel-Laplace transform, which makes it applicable to other problems. The method is applied to integrals with saddles. A powerful new method is presented to compute the Stokes multipliers. A numerical example is included.” The example demonstrates the power of the procedures developed within the paper. The basic example given is the equation as follows: \[ w^{(4)}(z)-3w^{(3)}(z)+ \Bigl(\textstyle{9\over 4} +\textstyle{1\over 2} z^{-2}\Bigr)w^{(2)}(z)-\Bigl(3 +\textstyle{3\over 4} z^{-2}\Bigr)w'(z)+\Bigl(\textstyle{5\over 4}+\textstyle {9\over 16}z^{-2}\Bigr) w(z)=0. \] As seen by the computations the results require the implementation of 47 terms of the asymptotic expansion of \(w_3 (z,n)\). The paper offers a significant new insight into this type of problem.

The method is based on the Borel-Laplace transform, which makes it applicable to other problems. The method is applied to integrals with saddles. A powerful new method is presented to compute the Stokes multipliers. A numerical example is included.” The example demonstrates the power of the procedures developed within the paper. The basic example given is the equation as follows: \[ w^{(4)}(z)-3w^{(3)}(z)+ \Bigl(\textstyle{9\over 4} +\textstyle{1\over 2} z^{-2}\Bigr)w^{(2)}(z)-\Bigl(3 +\textstyle{3\over 4} z^{-2}\Bigr)w'(z)+\Bigl(\textstyle{5\over 4}+\textstyle {9\over 16}z^{-2}\Bigr) w(z)=0. \] As seen by the computations the results require the implementation of 47 terms of the asymptotic expansion of \(w_3 (z,n)\). The paper offers a significant new insight into this type of problem.

Reviewer: J.Schmeelk (Richmond)

##### MSC:

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

34M99 | Ordinary differential equations in the complex domain |