  In recent work (math.QA/0309252) on multivariate hypergeometric integrals,
the author generalized a conjectural integral formula of van Diejen and
Spiridonov to a ten parameter integral provably invariant under an action of
the Weyl group E_7. In the present note, we consider the action of the affine
Weyl group, or more precisely, the recurrences satisfied by special cases of
the integral. These are of two flavors: linear recurrences that hold only up to
dimension 6, and three families of bilinear recurrences that hold in arbitrary
dimension, subject to a condition on the parameters. As a corollary, we find
that a codimension one special case of the integral is a tau function for the
elliptic Painlev\'e equation.

Classical Analysis and ODEs