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Abstract

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.

title = "The asymptotic expansion of a generalisation of the Euler-Jacobi series",

abstract = "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.",

author = "Paris, {Richard B.}",

year = "2016",

language = "English",

volume = "9",

pages = "3--18",

journal = "European Journal of Pure and Applied Mathematics",

Research output: Contribution to journal › Article

TY - JOUR

T1 - The asymptotic expansion of a generalisation of the Euler-Jacobi series

AU - Paris, Richard B.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

M3 - Article

VL - 9

SP - 3

EP - 18

JO - European Journal of Pure and Applied Mathematics

JF - European Journal of Pure and Applied Mathematics