Abstract
We prove that the maximal abelian extension tamely ramified at infinity of the rational function field over \(\mathbb {F}_q\) is generated by the values at the points in the algebraic closure of \(\mathbb {F}_q\) of the higher derivatives of the so-called Anderson and Thakur function \(\omega .\) We deduce a similar property for the special values of the higher derivatives of a new kind of \(L\)-series introduced by the second author.
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Acknowledgments
We warmly thank D. Goss and R. Perkins for fruitful discussions about the topics of this paper. We are very thankful to R. Perkins for having drawn our attention to a property similar to that of Corollary 1.2 on the values of \(\mathfrak {L}\) on \(\mathbb {F}_q^{ac}\). We heartily thank the referee for useful suggestions that helped us write a significantly improved text.
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F. Pellarin was supported by the contract ANR “HAMOT”, BLAN-0115-01.
