Noncommutative Extensions of Ramanujan’s 1ψ1 Summation ∗

Using functional equations, we derive noncommutative extensions of Ramanujan's 1 ψ 1 summation. 1. Introduction. Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have been the subject of recent study, see e.g. the papers by Duval and Ovsienko [DO], Grünbaum [G], Tirao [T], and some of the references mentioned therein. Of course, this subject is also closely related to the theory of orthogonal matrix polynomials which was initiated by Krein [K] and has experienced a steady development, see e.g. Durán and López-Rodríguez [DL]. Very recently, Tirao [T] considered a particular type of a matrix valued hyperge-ometric function (which, in our terminology, belongs to noncommutative hypergeo-metric series of " type I "). He showed, in particular, that the matrix valued hypergeo-metric function satisfies the matrix valued hypergeometric differential equation, and conversely that any solution of the latter is a matrix valued hypergeometric function. In [S2], the present author investigated hypergeometric and basic hypergeometric series involving noncommutative parameters and argument (short: noncommutative hypergeometric series, and noncommutative basic or Q-hypergeometric series) over a unital ring R (or, when considering nonterminating series, over a unital Banach algebra R) from a different, nevertheless completely elementary, point of view. These investigations were exclusively devoted to the derivation of summation formulae (which quite surprisingly even exist in the noncommutative case), aiming to build up a theory of explicit identities analogous to the rich theory of identities for hypergeometric and basic hypergeometric series in the classical, commutative case (cf. [Sl] and [GR]). Two closely related types of noncommmutative series, of " type I " and " type II " , were considered in [S2]. Most of the summations obtained there concern terminating series and were proved by induction. An exception are the noncommutative extensions of the nonterminating q-binomial theorem [S2, Th. 7.2] which were established using functional equations. Aside from the latter and some conjectured Q-Gauß summations, no other explicit summations for nonterminating noncommutative basic hypergeometric series were given. Furthermore, noncommutative bilateral basic hypergeometric series were not even considered. In this paper, we define noncommutative bilateral basic hypergeometric series of type I and type II (over an abstract unital Banach algebra R) and prove, using functional equations, noncommutative extensions of Ramanujan's 1 ψ 1 summation. These generalize the noncommutative Q-binomial theorem of [S2, Th. 7.2]. Our proof of the