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Exponentially small expansions associated with a generalised Mathieu series

R. B. Paris

We consider the generalised Mathieu series X∞ n=1 n γ (nλ + a λ) µ (µ > 0) when the parameters λ (> 0) and γ are both even integers for large complex a in the sector | arg 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.