Summation Formulae for Noncommutative Hypergeometric Series

Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have recently been studied by a number of researchers including (in alphabetical order) Durán, Duval, Grünbaum, Iliev, Ovsienko, Pacharoni, Tirao, and others. See [3, 6, 8, 9, 10, 11, 12, 16] for some selected papers. The subject of hypergeometric series involving matrices is closely related to and partly overlapping the theory of orthogonal matrix polynomials. The study of the latter was initiated by Krein [14] and subsequently has experienced a steady development. Whereas a good amount of theory of orthogonal matrix polynomials has already been worked out, see e.g. Durán and Van Assche [5], Durán and López-Rodŕıguez [4], and Tirao [16], it seems as appropriate to study noncommutative hypergeometric series (involving not only matrices but more generally arbitrary noncommutative parameters of some unit ring, or, in the case of infinite series, of some Banach algebra) from an entirely elementary point of view. This includes the search for identities for noncommutative hypergeometric and noncommutative basic hypergeometric series, extending their classical commutative versions which can be found, for instance, in the standard textbooks of Bailey [2], Slater [15], Gasper and Rahman [7], and of Andrews, Askey, and Roy [1]. This paper contains some results of our search which we hope will be the starting point of a systematic study towards a theory of identities for noncommutative hypergeometric series and their basic analogues (q-analogues). The special types of noncommutative hypergeometric series we are considering were inspired by a recent