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Équations fonctionnelles de Mahler et applications aux suites p-régulières

Abstract : p-Regular sequences, introduced by Allouche and Shallit, are a generalization of p-automatic sequences. The generating series of such a sequence can be either viewed as a formal power series, or as a holomorphic function (in the complex case); it satisfies a linear functional equation called "Mahler equation". In this work we give general results for these functional equations, that we then apply to the particular case of p-regular sequences. The formal framework is given in Chapters 1 to 3, where Mahlerian structures are studied. Chaper 4 shows transcendency of nonrational solutions through the study of their singularities. We thus extend a result which is well-known in the automatic case. Chapter 5 answers a question asked by Rubel by proving that, in one case, nonrational solutions are differentially transcendental (or hypertranscendental). Chapter 7, using well-known methods, builds on Chapter 4 to establish the transcendency of values of these functions, in connection with a question asked by Allouche and Shallit. Chapter 8 gives a very partial result in the direction of a conjecture by Loxton and van der Poorten. Chapter 6 sketches a study of the non-linear case.
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Contributor : Jean-Paul Allouche Connect in order to contact the contributor
Submitted on : Friday, August 7, 2015 - 11:05:47 AM
Last modification on : Saturday, December 4, 2021 - 3:42:55 AM
Long-term archiving on: : Wednesday, April 26, 2017 - 10:09:18 AM


  • HAL Id : tel-01183330, version 1



Bernard Randé. Équations fonctionnelles de Mahler et applications aux suites p-régulières. Mathématiques [math]. Université Bordeaux 1, 1992. Français. ⟨tel-01183330⟩



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