Hypergeometric functions for function fields and transcendence

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

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

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

Dinesh S. Thakur ¹ ; Zhi-Ying Wen ² ; Jia-Yan Yao ² ; Liang Zhao 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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     doi = {10.1016/j.crma.2009.03.006},
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Dinesh S. Thakur; Zhi-Ying Wen; Jia-Yan Yao; Liang Zhao. Hypergeometric functions for function fields and transcendence. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 467-472. doi : 10.1016/j.crma.2009.03.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.006/

Bibliographie
Cité par

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

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

[3] G.W. Anderson; W.D. Brownawell; M.A. Papanikolas 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] L. Denis Un critère de transcendance en caractéristique finie, J. Algebra, Volume 182 (1996), pp. 522-533

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Dinesh S. Thakur; Zhi-Ying Wen; Jia-Yan Yao; Liang Zhao Transcendence in positive characteristic and special values of hypergeometric functions, Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2011 (2011) no. 657 | DOI:10.1515/crelle.2011.055

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