NONCOMMUTATIVE HYPERGEOMETRIC AND BASIC HYPERGEOMETRIC EQUATIONS

Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100(14) (2003) 8138–8141] considered a matrix-valued analogue of the ₂F₁ Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao's result, extended to the more general setting of hypergeometric functions over an abstract unital Banach algebra. We provide a similar (but more complicated-looking) result for a second type of noncommutative ₂F₁ Gauß hypergeometric function. We further give q-analogues for both types of noncommutative hypergeometric equations.