Comptes Rendus
no. 11
Some transcendental functions over function fields with positive characteristic
[Certaines fonctions transcendantes sur des corps de fonctions de caractéristique positive]
Jia-Yan Yao1
1 Department of Mathematics, Nonlinear Science Center, Wuhan University, Wuhan 430072, People's Republic of China
Comptes Rendus. Mathématique, Volume 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
Jia-Yan Yao 1

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

[1] J.-P. Allouche Somme des chiffres et transcendance, Bull. Soc. Math. France, Volume 110 (1982), pp. 279-285

[2] J.-P. Allouche Automates finis en théorie des nombres, Exposition. Math., Volume 5 (1987), pp. 239-266

[3] J.-P. Allouche Note sur un article de Sharif et Woodcock, Sém. Théor. Nombres Bordeaux, Volume 1 (1989), pp. 163-187

[4] J.-P. Allouche Sur la transcendance de la série formelle Π, Sém. Théor. Nombres Bordeaux, Volume 2 (1990), pp. 103-117

[5] J.-P. Allouche Transcendence of the Carlitz–Goss gamma function at rational arguments, J. Number Theory, Volume 60 (1996), pp. 318-328

[6] C. Cadic, Interprétation p-automatique des groupes formels de Lubin–Tate et des modules de Drinfeld réduits, Thèse, Université de Limoge, 1999, http://www.unilim.fr/laco/theses/1999/T1999_01.ps

[7] L. Carlitz On certain functions connected with polynomials in a Galois field, Duke Math. J., Volume 1 (1935), pp. 137-168

[8] G. Christol Ensembles presque périodiques k-reconnaissables, Theoret. Comput. Sci., Volume 9 (1979), pp. 141-145

[9] G. Christol; T. Kamae; M. Mendès France; G. Rauzy Suites algébriques, automates et substitutions, Bull. Soc. Math. France, Volume 108 (1980), pp. 401-419

[10] D. Goss Basic Structures of Function Field Arithmetic, Springer, 1998

[11] T. Harase Algebraic elements in formal power series rings, Israel J. Math., Volume 63 (1988), pp. 281-288

[12] M. Mendès France; J.-Y. Yao Transcendence and the Carlitz–Goss gamma function, J. Number Theory, Volume 63 (1997), pp. 396-402

[13] H. Sharif; C.F. Woodcock Algebraic functions over a field of positive characteristic and Hadamard products, J. London Math. Soc., Volume 37 (1988), pp. 395-403

[14] D.S. Thakur Automata and transcendence, Number Theory, Tiruchirapalli, 1996, Contemp. Math., 210, American Mathematical Society, Providence, RI, 1998, pp. 387-399

[15] L.I. Wade Certain quantities transcendental over GF(pn,x), Duke Math. J., Volume 8 (1941), pp. 701-720

[16] L.I. Wade Certain quantities transcendental over GF(pn,x), II, Duke Math. J., Volume 10 (1943), pp. 587-594

[17] L.I. Wade Remarks on the Carlitz ψ-functions, Duke Math. J., Volume 13 (1946), pp. 71-78

[18] L.I. Wade Transcendence properties of the Carlitz ψ-functions, Duke Math. J., Volume 13 (1946), pp. 79-85

[19] M. Waldschmidt Transcendence problems connected with Drinfeld modules, Istanbul Üniv. Fen Fak. Mat. Derg., Volume 49 (1990), pp. 57-75

[20] Z.-Y. Wen; J.-Y. Yao Transcendence, automata theory and gamma functions for polynomial rings, Acta Arith., Volume 101 (2002), pp. 39-51

[21] J.-Y. Yao, Contribution à l'étude des automates finis, Thèse, Université de Bordeaux I, 1996

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