Exponentially small expansions associated with a generalised Mathieu series

الأصل · EN

We consider the generalised Mathieu series ∑ₙ₌₁∞ nγ/(nλ+aλ)μ (μ>0) when the parameters λ (>0) and γ are even integers for large complex a in the sector |a|<π/λ. The asymptotics in this case consist of a 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.