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The asymptotic expansion of a generalisation of the Euler-Jacobi series. (English) Zbl 1362.30043

Summary: We consider the asymptotic expansion of the sum \[ S_p(a;w)=\sum^\infty_{n=1}\frac{e^{-an^p}}{n^w} \] as \(a\to0\) in \(|\arg a|<\frac12\pi\) 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 \(S_p(a;w)\) 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.

MSC:

30E15 Asymptotic representations in the complex plane
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References:

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