A NEW An EXTENSION OF RAMANUJAN'S 1 1 SUMMATION WITH APPLICATIONS TO MULTILATERAL An SERIES

Abstract. In this article, we derive some identities for multilateral basic hypergeometric series associated to the root system An. First, we apply Ismail’s [15] argument to an An q-binomial theorem of Milne [25, Th. 5.42] and derive a new An generalization of Ramanujan’s 1ψ1 summation theorem. From this new An 1ψ1 summation and from an An 1ψ1 summation of Gustafson [9] we deduce two lemmas for deriving simple An generalizations of bilateral basic hypergeometric series identities. These lemmas are closely related to the Macdonald identities for An. As samples for possible applications of these lemmas, we provide several An extensions of Bailey’s 2ψ2 transformations, and several An extensions of a particular 2ψ2 summation.