This will be a very introductory talk on the computational theory of special function transformations. The special functions considered will be the Gauss hypergeometric function, known as ₂F₁, and the Heun function. They satisfy second-order differential equations on the complex projective line (i.e., the Riemann sphere), with three and four singular points, respectively. Abstractly, these equations specify flat meromorphic connections on rank-2 vector bundles over the projective line. The special function transformations to be discussed can be viewed as automorphisms of these bundles-with-connection. But they will be approached very concretely indeed, as transformations of power series in a single variable. Although the first such transformations discovered, for the Gauss function, were worked out by Kummer in 1836, the Heun function is only now being treated. Many of our results have appeared in R. S. Maier, "The 192 solutions of the Heun equation", Mathematics of Computation 76, 811-843 (2007).

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