An identity in combinatorial analysis

Abstract

In a very recent paper [1], Basil Gordon discusses generalizations of Jacobi's identitywhere x and z are complex numbers and |x| <1. He notes that some of its consequences, inter alia Euler's formulaare of interest in number theory and combinatory analysis. He proves the apparently new and striking resultwhere |s|<1, and also considers the possibility of generalizations. His methods are algebraic and quite simple, but perhaps do not make obvious what underlies such formulae. It may be worth while to do so, especially since the details become simpler and the presentation more perspicuous. The method given here assumes no more knowledge than his does, although the new proof is expressed in terms of theta-functions, in simple properties of which, formulae such as (3) have their origin. Further, (3) appears in a slightly more symmetrical form.