Truncation and manipulation of multivariate power series

Although much has been published on the manipulation of power series in a single variable, there is little in the literature on the topic of performing operations on multivariate power series. This paper initially concerns a novel prescription for the truncation of multivariate power series, namely weighted truncation. This prescription for truncation allows the retention of terms up to different degrees in the different variables. In practice, this feature is important for efficiency. Subsequently, a variety of routines are developed for operating on power series which have been truncated in this fashion. These operations include the determination of products, quotients and special functions (e.g. logarithms, trigonometric functions, etc.) of multivariate power series, as well as procedures for composition and reversion.