On certain generating functions in positive characteristic

TL;DR

This paper introduces new methods to analyze generating functions in positive characteristic, revealing their connections to deformations of the Carlitz logarithm, special functions, and L-functions, with implications for understanding their functional identities and specializations.

Contribution

It develops novel techniques for studying generating functions in positive characteristic, linking them to Carlitz deformations and Gauss-Thakur sums, and establishing explicit functional identities.

Findings

Established explicit functional identities involving Carlitz and Anderson-Thakur functions.

Linked generating functions to deformations of the Carlitz logarithm and L-functions.

Connected specializations at roots of unity to Gauss-Thakur sums.

Abstract

We present new methods for 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 logarithm 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 introduced by the first author. This paper also deals with specializations at roots of unity of these generating functions, producing a link with Gauss-Thakur sums.