On certain generating functions in positive characteristic

Federico Pellarin (1) , Rudolph Perkins (1)

1 ICJ - Institut Camille Jordan

We introduce new methods into the study of a class of generating functions introduced by the second author which carry some formal similarities with the Hurwitz zeta function. We prove functional identities which establish an explicit connection with certain deformations of the Carlitz logarithms introduced by M. Papanikolas and involve the Anderson-Thakur function and the Carlitz exponential function. They collect certain functional identities in families for a new class of $L$-functions in equal positive characteristic introduced by the first author. This paper also deals with specializations at roots of unity of these functions, a link with Gauss-Thakur sums, and an analogue of the classical Carlson theorem from complex analysis.

Mots clés: Positive characteristic; Carlitz module; Anderson generating functions; $L$-series; special values; periodic functions

Domaines: Théorie des nombres [math.NT]

Federico Pellarin, Rudolph Perkins. On certain generating functions in positive characteristic. 2014. ⟨hal-01066478v1⟩