The Lerch Zeta function III. Polylogarithms and special values

This paper studies algebraic and analytic structures associated with the Lerch zeta function, complex variables viewpoint taken in part II. The Lerch transcendent Φ(s,z,c):=∑n=0∞zn(n+c)s is obtained from the Lerch zeta function ζ(s, a, c) by the change of variable z= e^{2 π i a}. 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× (P¹(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).