The asymptotic expansion of a generalisation of the Euler-Jacobi series

We consider the asymptotic expansion of the sum Sp(a; w) = ∞Ʃ n=1 e−anp/nw as a → 0 in | arg a| <1/2π for arbitrary finite p > 0 and w > 0. Our attention is concentrated mainly on the case when p and w are both even integers, where the expansion consists of a finite algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for Sp (a; w) analogous to the well-known Poisson-Jacobi transformation for the sum with p = 2 and w = 0. Numerical results are given to illustrate the accuracy of the expansion obtained.