On the $L$-series of F. Pellarin

The calculation, by L.\ Euler, of the values at positive even integers of the Riemann zeta function, in terms of powers of $\pi$ and rational numbers, was a watershed event in the history of number theory and classical analysis. Since then many important analogs involving $L$-values and periods have been obtained. In analysis in finite characteristic, a version of Euler’s result was given by L.\ Carlitz \cite{ca2} in the 1930’s which involved the period of a rank 1 Drinfeld module (the Carlitz module) in place of $\pi$. In a very original work \cite{pe2}, F.\ Pellarin has quite recently established a “deformation” of Carlitz’s result involving certain $L$-series and the deformation of the Carlitz period given in \cite{at1}. Pellarin works only with the values of this $L$-series at positive integral points. We show here how the techniques of \cite{go1} also allow these new $L$-series to be analytically continued – with associated trivial zeroes – and interpolated at finite primes.