Holomorphic solutions of E-operators

Abstract

We solve the problem of describing the solutions of E-operators of order μ ≥ 1 admitting at z = 0 a basis over C of local solutions which are all holomorphic at z = 0. We prove that the components of such a basis can be taken of the form $\sum {_{j = 1}^\ell } {P_j}\left( z \right){e^{{\beta _{{j^z}}}}}$, where ℓ ≤ μ, β ₁,...,β _ℓ ∈ $\overline {\mathbb{Q}}$ ^x, and P ₁(z),..., P _ℓ(z) ∈ $\overline {\mathbb{Q}}$[z].