Transformation Formulas and Rational Representations for Hypergeometric Functions
The univariate special functions of classical analysis include the Gauss hypergeometric function and its generalizations. They are defined as convergent Maclaurin series, or alternatively as solutions of ODEs in the complex domain. They satisfy many transformation and evaluation identities, which can be derived by series manipulation and other classical techniques. But the ODE interpretation may be deeper. Hypergeometric ODEs have three singular points on the Riemann sphere (x=0,1,infinity), and therefore define flat connections on vector bundles over the triply punctured sphere, each puncture having a monodromy matrix. In this talk we explain some of our results, including (i) the uniformization of certain hypergeometric functions that are algebraic (i.e., have a finite monodromy group), and (ii) the ladders of quadratic and cubic transformation formulas that relate certain non-algebraic hypergeometric functions.