An unsolvable Cousin problem

Abstract

We construct a Laurent series ∑ n = − ∞ ∞ A n z n \sum \nolimits _{n = - \infty }^\infty {{A_n}{z^n}} , convergent in an annulus { ρ > | z | > 1 / ρ } \left \{ {\rho > \left | z \right | > 1/\rho } \right \} for some ρ > 1 \rho > 1 , which satisfies an algebraic differential equation (ADE) there, but such that neither of the series ∑ n = 0 ∞ A n z n \sum \nolimits _{n = 0}^\infty {{A_n}{z^n}} and ∑ n = − ∞ − 1 A n z n \sum \nolimits _{n = - \infty }^{ - 1} {{A_n}{z^n}} satisfies any ADE.