广义Mathieu级数的指数小展开式
Exponentially small expansions associated with a generalised Mathieu series
摘要：
我们考虑广义Mathieu系列\[\sum_{n=1}^\infty \frac{n^γ}{(n^λ+a^λ)^μ}\qquad (μ>0)\]时，参数$λ$（$>0$）和$γ$是偶数的大复杂$a$的部门$|\arg\,a|<π/λ$。在这种情况下的渐近性包括一个{ \it有限}代数展开以及一个无穷序列越来越次显性指数小的扩展。当$μ$也是一个正整数时，可以给出这个级数的封闭形式的求值。数值结果说明了所得到的展开式的精确性。
英文摘要：
We consider the generalised Mathieu series \[\sum_{n=1}^\infty \frac{n^γ}{(n^λ+a^λ)^μ}\qquad (μ>0)\] when the parameters $λ$ ($>0$) and $γ$ are even integers for large complex $a$ in the sector $|\arg\,a|<π/λ$. The asymptotics in this case consist of a {\it finite} algebraic expansion together with an infinite sequence of increasingly subdominant exponentially small expansions. When $μ$ is also a positive integer it is possible to give closed-form evaluations of this series. Numerical results are given to illustrate the accuracy of the expansion obtained.
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