An infinite family of engel expansions of Rogers-Ramanujan type

Abstract

The extended Engel expansion is an algorithm that leads to unique series expansions of q-series. Various examples related to classical partition theorems, including the Rogers-Ramanujan identities, have been given recently. The object of this paper is to show that the new and elegant Rogers-Ramanujan generalization found by Garrett, Ismail, and Stanton also fits into this framework. This not only reveals the existence of an infinite, parameterized family of extended Engel expansions, but also provides an alternative proof of the Garrett, Ismail, and Stanton result. A finite version of it, which finds an elementary proof, is derived as a by-product of the Engel approach. © 2000 Academic Press.