Convolutions of Goss and Pellarin $L$-series

TL;DR

This paper extends the understanding of special values of convoluted Goss and Pellarin $L$-series for Drinfeld modules, connecting them to Stark units and generalizing previous formulas to higher ranks.

Contribution

It introduces new special value formulas for convolutions of Goss and Pellarin $L$-series for arbitrary rank Drinfeld modules, expanding prior results beyond the Carlitz case.

Findings

Extended special value formulas to higher rank Drinfeld modules.

Connected convolution $L$-series with Stark units.

Expressed identities via Schur polynomial specializations.

Abstract

We establish special value results of convolutions of Goss and Pellarin $L$-series attached to Drinfeld modules that take values in Tate algebras. Applying the class module formula of Demeslay to certain rigid analytic twists of one Drinfeld module by another, we extend the special value formula for the Pellarin $L$-function associated to the Carlitz module and the Anderson-Thakur function to Drinfeld modules of arbitrary rank and their rigid analytic trivializations. By way of the theory of Schur polynomials these identities take the form of specializations of convolutions of Rankin-Selberg type. These convolution $L$-series are also identified with covolumes of Stark units.