  A generalized Tate curve is a universal family of curves with fixed genus and
degeneration data which becomes Schottky uniformized Riemann surfaces and
Mumford curves by specializing moduli and deformation parameters. By
considering each generalized Tate curve as a family of degenerating Riemann
surfaces, we give explicit formulas of the period isomorphism between its de
Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin
connection, variation of Hodge structure and monodromy weight filtration. A
remarkable fact is that similar formulas hold also for families of Mumford
curves. Furthermore, we show that for a generalized Tate with maximally
degenerate closed fiber, its local unipotent periods can be expressed as power
series in the deformation parameters whose coefficients are multiple
polylogarithm functions. This p-adic version is also given.
