### Abstract

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× (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).

Original language | English (US) |
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Article number | 2 |

Journal | Research in Mathematical Sciences |

Volume | 3 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Mathematics (miscellaneous)
- Computational Mathematics
- Applied Mathematics

### Cite this

*Research in Mathematical Sciences*,

*3*(1), [2]. https://doi.org/10.1186/s40687-015-0049-2