Holomorphic solutions of \(E\)-operators. (English) Zbl 1430.11103
Summary: We solve the problem of describing the solutions of \(E\)-operators of order \(\mu\geq 1\) admitting at \(z = 0\) a basis over \(\mathbb C\) 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 \(\sum_{j = 1}^\ell P_j(z)e^{\beta_{j^z}}\), where \(\ell\leq \mu, \beta_{1},\dotsc, \beta_\ell \in \overline {\mathbb{Q}}^{\times}\), and \(P_{1}(z),\dotsc, P_\ell(z)\in\overline {\mathbb{Q}}[z]\).
MSC:
| 11J91 | Transcendence theory of other special functions |
| 30B10 | Power series (including lacunary series) in one complex variable |
| 33E30 | Other functions coming from differential, difference and integral equations |
References:
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