Noncommutative hypergeometric and basic hypergeometric equations
J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138--8141]
considered a matrix-valued analogue of the
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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