Generalized multiple Dirichlet series and generalized multiple polylogarithms

where t is a complex variable. By Assumption I, we see that the series (1.3) is convergent when <t < 0. We further assume the following: (Assumption II ) ψ(s) can be continued analytically to the whole complex plane C, and holomorphic for all s ∈ C. In any fixed strip σ1 ≤ σ ≤ σ2, ψ(s; u) is uniformly convergent to ψ(s) as u → 1 + 0. Furthermore there exists a certain θ0 = θ0(σ1, σ2) ∈ R with 0 ≤ θ0 < π/2 such that ψ(s;u) = O ( eθ0|τ | ) as |τ | → ∞.