Alessandro Conflitti and Michael J. Schlosser

Noncommutative hypergeometric and basic hypergeometric equations

(16 pages)

Abstract. 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.

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