On Multiple Asymptotic Expansions

In previous papers asymptotic meaning has been attached to the formal double sums $\sum _{m = 0}^\infty \sum _{n = 0}^\infty a_{mn} x^{ - n} e^{ - \lambda mx} (\lambda > 0)$ as $x \to 0$ in a sector of the right half-plane. Formal applications of these sums to elastic scattering and to singular perturbation problems have also been given. In this paper the concept of asymptotic sequences and series are extended to define more general double sums. Multiplication and differentiation properties of these sequences and series are investigated. Application of these more general asymptotic expansions to systems of first order, linear ordinary differential equations is presented.