This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent (s, z, c) := ∞ n=0 z n (n+c) s is obtained from the Lerch zeta function ζ (s, a, c) by the change of variable z = e 2πia . We show that it analytically continues to a maximal domain of holomorphy in three complex variables (s, z, c), as a multivalued function defined over the base manifold C × (P 1 (C) {0, 1, ∞}) × (C Z) and compute the monodromy functions describing the multivaluedness. For positive integer values s = m and c = 1 this function is closely related to the classical m-th order polylogarithm Li m (z). We study its behavior as a function of two variables (z, c) for "special values" where s = m is an integer. For m ≥ 1 we show that it is a one-parameter deformation of Li m (z), which satisfies a linear ODE, depending on c ∈ C, of order m + 1 of Fuchsian type on the Riemann sphere. We determine the associated (m + 1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of Li m (z).