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Transcendence and the Carlitz–Goss Gamma Function

✍ Scribed by Michel Mendès France; Jia-yan Yao

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
285 KB
Volume
63
Category
Article
ISSN
0022-314X

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✦ Synopsis

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 the values of the Carlitz Goss gamma function. This concludes the transcendence study of the values of this function initiated and developed by D. Thakur and pursued by A. Thiery, J. Yu, and L. Denis and J.-P. Allouche. The proof of our result, as that of J.-P. Allouche, uses derivation of formal power series and the theorem of G. Christol, T. Kamae, M. MendeÁ s France, and G. Rauzy.

1997 Academic Press

Now if n is a p-adic integer, i.e., an element of Z p , we write n= : j=0 n j q j , with 0 n j <q, article no. NT972104 396 0022-314XÂ97 25.00

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