A hypergeometric function is a solution to

(*) w'' + p(z) w' + q(z) w = 0

where q has at most simple poles and q has at most double poles at 0,1,infty.

That differential equation is equivalent to the data of a flat connection D = D(p,q) on the trivial vector bundle of rank 2 over C - {0,1}, with regular singularities at 0,1,infty. If w is a local solution to (*), then (w,w') is a section of the trivial bundle with D(w,w') = 0.

Hypergeometric functions satisfy lots of identities, usually involving two or more different equations of the form (*). Is there a way to package these identities, or some fraction of them, as an interesting algebraic structure on the collection of such differential equations, or on the collection of flat connections on the trivial bundle?


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