Hypergeometric functions for function fields and transcendence

Thakur, Dinesh S.; Wen, Zhi-Ying; Yao, Jia-Yan; Zhao, Liang

doi:10.1016/j.crma.2009.03.006

Number Theory

Hypergeometric functions for function fields and transcendence
[Fonctions hypergéométriques pour les corps de fonctions et transcendance]

Présenté par : Jean-Pierre Kahane

Thakur, Dinesh S.¹ ; Wen, Zhi-Ying ² ; Yao, Jia-Yan ² ; Zhao, Liang ²

¹ Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
² Department of Mathematics, Tsinghua University, Beijing 100084, PR China

Comptes Rendus. Mathématique, Tome 347 (2009) no. 9-10, pp. 467-472.

Résumé
Abstract

Nous étudions les fonctions hypergéométriques pour $F_{q} [T]$ , et démontrons dans le cas entier (non polynomial) la transcendance de leurs valeurs spéciales aux arguments algébriques non nuls qui engendrent des extensions du corps de fonctions rationnelles avec au plus $q - 1$ places à l'infini. Nous caractérisons aussi dans le cas équilibré l'algébricité des fonctions hypergéométriques.

We study hypergeometric functions for $F_{q} [T]$ , and show in the entire (non-polynomial) case the transcendence of their special values at nonzero algebraic arguments which generate extension of the rational function field with less than q places at infinity. We also characterize in the balanced case the algebraicity of hypergeometric functions.

Reçu le : 2008-12-23
Accepté le : 2009-03-02
Publié le : 2009-04-03

DOI : 10.1016/j.crma.2009.03.006

Affiliations des auteurs :

Thakur, Dinesh S. ¹ ; Wen, Zhi-Ying ² ; Yao, Jia-Yan ² ; Zhao, Liang ²

¹ Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA
² Department of Mathematics, Tsinghua University, Beijing 100084, PR China

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     title = {Hypergeometric functions for function fields and transcendence},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {467--472},
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Thakur, Dinesh S.; Wen, Zhi-Ying; Yao, Jia-Yan; Zhao, Liang. Hypergeometric functions for function fields and transcendence. Comptes Rendus. Mathématique, Tome 347 (2009) no. 9-10, pp. 467-472. doi : 10.1016/j.crma.2009.03.006. http://www.numdam.org/articles/10.1016/j.crma.2009.03.006/

Bibliographie
Cité par

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

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

[3] Anderson, G.W.; Brownawell, W.D.; Papanikolas, M.A. Determination of the algebraic relations among special Γ-values in positive characteristic, Ann. Math., Volume 160 (2004) no. 2, pp. 237-313

[4] C.-Y. Chang, M. Papanikolas, D. Thakur, J. Yu, Algebraic independence of arithmetic gamma values and Carlitz zeta values, (2008), submitted for publication

[5] L. Denis, Transcendance et dérivées de l'exponentielle de Carlitz, in: S. David (Ed.), Progr. Math., vol. 116, Séminaire de Théorie des Nombres (Paris, 1991–92), Birkhäuser, 1993, pp. 1–21

[6] Denis, L. Un critère de transcendance en caractéristique finie, J. Algebra, Volume 182 (1996), pp. 522-533

[7] Denis, L. Valeurs transcendantes des fonctions de Bessel–Carlitz, Ark. Mat., Volume 36 (1998), pp. 73-85

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

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

[10] Laohakosol, V.; Kongsakorn, K.; Ubolsri, P. Some transcendental elements in positive characteristic, Sci. Asia, Volume 26 (2000), pp. 39-48

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

[12] Papanikolas, M. Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., Volume 171 (2008), pp. 123-174

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

[14] Thakur, D.S. Hypergeometric functions for function fields, Finite Fields Appl., Volume 1 (1995), pp. 219-231

[15] Thakur, D.S. Transcendence of gamma values for $F_{q} [T]$ , Ann. Math., Volume 144 (1996) no. 2, pp. 181-188

[16] Thakur, D.S. An alternate approach to solitons for $F_{q} [T]$ , J. Number Theory, Volume 76 (1999), pp. 301-319

[17] Thakur, D.S. Hypergeometric functions for function fields II, J. Ramanujan Math. Soc., Volume 15 (2000), pp. 43-52

[18] Thakur, D.S. Integrable systems and number theory in finite characteristic, Adv. Nonlinear Math. Sci. Physica D, Volume 152/153 (2001), pp. 1-8

[19] Thakur, D.S. Function Field Arithmetic, World Scientific Publishing Co., Inc., 2004

[20] Villegas, F.R. Integral ratios of factorials and algebraic hypergeometric functions | arXiv

[21] Wade, L.I. Certain quantities transcendental over $G F (p^{n}, x)$ , Duke Math. J., Volume 8 (1941), pp. 701-720

[22] Wade, L.I. Certain quantities transcendental over $G F (p^{n}, x)$ , II, Duke Math. J., Volume 10 (1943), pp. 587-594

[23] Wade, L.I. Two types of function field transcendental numbers, Duke Math. J., Volume 11 (1944), pp. 755-758

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

[25] Yao, J.-Y. A transcendence criterion in positive characteristic and applications, C. R. Acad. Sci. Paris, Sér. I, Volume 343 (2006), pp. 699-704

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