Kronecker limit formula over global function fields

The aim of this paper is to present a function field analogue of the classical Kronecker limit formula. We first introduce a ``non-holomorphic'' Eisenstein series on the Drinfeld half plane, and connect its ``second term'' with Gekeler's discriminant function. One application is to express the Taguchi height of rank $2$ Drinfeld modules with complex multiplication in terms of the logarithmic derivative of the corresponding zeta functions. Moreover, from the integral form of the Rankin-type $L$-function associated to two ``Drinfeld-type'' newforms, we then derive a formula for a non-central special derivative of the $L$-function in question. Adapting the classical approach, we also obtain a Kronecker-type solution for Pell's equation over function fields.