**Inclusion relations among methods of four-dimensional summability compounded from given four-dimensional methods **

* Richard F. Patterson *

**Abstract**

The goals of this paper include the introduction of a new four-dimensional summability method construction by compounding a single four-dimensional method. The examination of this method begins with the characterization of its RH-regularity properties. In addition, the following inclusion and consistent theorems will be presented.

If *α*_{ m }}, {*β*_{ n }} are sequences such that {*α*_{ m }} and {*β*_{ n }} are monotone increasing with *m* and *n* and if the transformations *B*(*α*_{ m },*β*_{ n }) are factorable and RH-regular then *B*(*α*_{ m },*β*_{ n }).

The RH-regular matrix transformations of the form *B*(*r*_{ m },*s*_{ n }) for which *r*_{1}≤*r*_{2}≤*r*_{3}≤⋯ and *s*_{1}≤*s*_{2}≤*s*_{3}≤⋯ constitute a double sequence of consistent family. Other implications and variations will also be presented.