The need for a new classification of double hypergeometric series

Abstract

An elementary proof is given of a linear transformation which changes a particular double hypergeometric series of order two into a series of order three. A similar transformation holds for a particular series of arbitrary order in two or more variables. The change in order provides new evidence, the most compelling to date, that the order of such a series is not a fundamental property. This conclusion undermines the accepted classification of hypergeometric series in more than one variable.