Combinatorial proof of the transcendence of $L(1,\ddot{I}{‡}_s)/\hat{I}$

Abstract

We give a combinatorial proof of the transcendence of $L(1,\ddot{I}{‡}_s)/\hat{I}$, where $L(1,\ddot{I}{‡}_s)$ (resp. $\hat{I}$) is the analogue in characteristic $p$ of the function $L$ of Dirichlet (resp. $\ddot{I}€$). This result has been proven by G. Damamme using the criteria of de Mathan. Our proof is based on the Theorem of Christol and another property of $k$-automatic sequences.