Eñe product in the transalgebraic class

Abstract: We define transalgebraic functions on a compact Riemann surface as meromorphic functions except at a finite number of punctures where they have finite order exponential singularities. This transalgebraic class is a topological multiplicative group. We extend the action of the e~ne product to the transcendental class on the Riemann sphere. This transalgebraic class, modulo constant functions, is a commutative ring for the multiplication, as the additive structure, and the e~ne product, as the multiplicative structure. In particular, the divisor action of the e~ne product by multiplicative convolution extends to these transalgebraic divisors. The polylogarithm hierarchy appears related to transalgebraic e~ne poles of higher order.