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The Asymptotic Expansion of a Generalisation of the Euler-Jacobi Series

Journal Name:

  • European Journal of Pure and Applied Mathematics

Publication Year:

  • 2016

Key Words:

Author Name
Abstract (2. Language): 
We consider the asymptotic expansion of the sum Sp(a;w) = ∞X n=1 e−anp nw as a →0 in |arg a| < 1 2 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.
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