A new application of Lagrange-Bürmann expansions. I. General principle

A new scheme is developed for improving the convergence of slowly convergent series solutions. The method is based on a transformation of variables of similarity form whereby the resulting composite function is constructed by its Lagrange-Bürmann expansion. It is the improved convergence of the new expansion that we take most advantage of in this method. The convergence of the Lagrange-Bürmann expansion as well as its inversion scheme is proved for analytic (object) functions. The inversion is required to recover from the Lagrange-Bürmann expansion the object function which is imbedded in the mapping functions. Several numerical examples demonstrate the improved convergence of the new method. The improvement owes much to the invariance properties of the mapping function under a group and the “built-in” feature of analytic continuation of the method. These features are elucidated in detail.