Some transcendental functions over function fields with positive characteristic
[Certaines fonctions transcendantes sur des corps de fonctions de caractéristique positive]
Yao, Jia-Yan1
1 Department of Mathematics, Nonlinear Science Center, Wuhan University, Wuhan 430072, People's Republic of China
Comptes Rendus. Mathématique, Tome 334 (2002) no. 11, pp. 939-943.

Dans ce travail, nous allons définir deux familles de fonctions sur des corps de fonctions de caractéristique positive et montrer qu'une telle fonction est transcendante si et seulement si sa suite génératrice n'est pas ultimement nulle. Comme conséquence, l'exponentielle de Carlitz et le logarithme de Carlitz sont des fonctions transcendantes. Notre preuve est élémentaire dans le sens que nous allons utiliser seulement un théorème dû à H. Sharif et C. Woodcock, ainsi qu'à T. Harase qui généralise le théorème de Christol pour les suites automatiques.

In this work we shall define two families of functions over function fields with positive characteristic and show that such a function is transcendental if and only if its generating sequence is not ultimately zero. As a result, the Carlitz exponential and the Carlitz logarithm are transcendental functions. Our proof is elementary in the sense that we only use a theorem due to H. Sharif and C. Woodcock, and to T. Harase which generalizes the theorem of Christol about automatic sequences.

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DOI : 10.1016/S1631-073X(02)02378-6
Yao, Jia-Yan 1

1 Department of Mathematics, Nonlinear Science Center, Wuhan University, Wuhan 430072, People's Republic of China 
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Yao, Jia-Yan. Some transcendental functions over function fields with positive characteristic. Comptes Rendus. Mathématique, Tome 334 (2002) no. 11, pp. 939-943. doi : 10.1016/S1631-073X(02)02378-6. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02378-6/

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