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:
[1] | Y. André, G-functions and Geometry, Aspects of Mathematics, Vol. E13, Friedr. Vieweg & Sohn, Braunschweig, 1989. · Zbl 0561.10016 |
[2] | Y. André, Séries Gevrey de type arithmétique I. Théorèmes de pureté et de dualité, Annals of Mathematics 151 (2000), 705-740. · Zbl 1037.11049 · doi:10.2307/121045 |
[3] | Y. André, Séries Gevrey de type arithmétique II. Transcendance sans transcendance, Annals of Mathematics 151 (2000), 741-756. · Zbl 1037.11050 · doi:10.2307/121046 |
[4] | Bertrand, D., Le théorème de Siegel-Shidlovsky revisité, in Number Theory, 51-67 (2012) · Zbl 1282.11081 |
[5] | G. V. Chudnovsky, On applications of diophantine approximations, Proceedings of the National Academy of Sciences of the United States of America 81 (1984), 7261-7265. · Zbl 0566.10029 · doi:10.1073/pnas.81.22.7261 |
[6] | D. V. Chudnovsky and G. V. Chudnovsky, Applications of Padé approximations to Diophantine inequalities in values of G-functions, in Number Theory (New York, 1983/84), Lecture Notes in Mathematics, Vol. 1135, Springer, Berlin, 1985, pp. 9-51. · Zbl 0561.10016 |
[7] | V. Ditkine and A. Proudnikov, Calcul Opérationnel, Mir, Moscow, 1979. · Zbl 0494.44001 |
[8] | B. Dwork, G. Gerrotto and F. J. Sullivan, An Introduction to G-functions, Annals of Mathematical Studies, Vol. 133, Princeton University Press, Princeton, NJ, 1994. · Zbl 1370.11090 |
[9] | S. Fischler and T. Rivoal, Arithmetic theory of E-operators, Journal de l’École polytechnique. Mathématiques 3 (2016), 31-65. · Zbl 1370.11090 · doi:10.5802/jep.28 |
[10] | A. B. Shidlovskii, Transcendental Numbers, de Gruyter Studies in Mathematics, Vol. 12, Walter de Gruyter, Berlin, 1989. · Zbl 0689.10043 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.