Abstract
The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series; 2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians; 3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation; 4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.
Highlights
This paper is a direct continuation of [1] [2] and contains some complements to the theory developed therein with the purpose of completing the general theory
The obtained results are quite natural and are based on the asymptotic study of a Vandermonde determinant with a gap in the exponents, a study which parallels the analogous investigation for standard Vandermondians in the previous papers. §4 contains a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation
The theory developed in [1] [2] is necessarily based on asymptotic differential equations in order to define higher-order types of variation for differentiable functions, whereas the more general Karamata theory is based on asymptotic functional equations
Summary
This paper is a direct continuation of [1] [2] and contains some complements to the theory developed therein with the purpose of completing the general theory. This first section, besides a list of notations, contains a summary of the various involved classes of functions and their main characterizations
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