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Transcendence of the Carlitz–Goss Gamma Function at Rational Arguments

✍ Scribed by Jean-Paul Allouche

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
488 KB
Volume
60
Category
Article
ISSN
0022-314X

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✦ Synopsis

We show that the values of the Carlitz Goss Gamma function for F q [X] are transcendental over F q (X) for all arguments which are rational and not in N. We also show the transcendence of monomials built on the values of the Carlitz Goss Gamma function. This generalizes previous partial results due to Thakur, Thiery, Yu, and Denis. Our proof uses derivation of formal power series and the theorem of Christol, Kamae, MendeÁ s France, and Rauzy.

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Transcendence and the Carlitz–Goss Gamma
Transcendence and the Carlitz–Goss Gamma Function
✍ Michel Mendès France; Jia-yan Yao 📂 Article 📅 1997 🏛 Elsevier Science 🌐 English ⚖ 285 KB

In this article, we show that a value of the Carlitz Goss gamma function for the ring F q [X] is transcendental over the field F q (X) if and only if the argument is not an element of N=[0, 1, 2, . . .]. We also answer a question of J.-P. Allouche, and we show the transcendence of monomials built on