The Heun equation is the canonical second-order differential equation on
the Riemann sphere with four regular singular points, just as the Gauss
hypergeometric equation (GHE) is the canonical equation with three. The
GHE has many symmetries, which generate (1) Kummer's well-known 24 series
solutions of the GHE, (2) `contiguity relations' relating solutions of
different GHEs, and (3) an explicit solution of the GHE connection problem.
In this talk, we explain how analogous results on the Heun equation can be
derived. For instance, the Heun equation has a family of 192 series
solutions, analogous to Kummer's 24. Its contiguity relations are based on
Schlesinger transformations, which can be viewed alternatively as the
source of known Bäcklund symmetries of the sixth equation of Painlevé; but
which have not been worked out systematically, until now.
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