Two Generator Subgroups ofSL(2,C) and the Hypergeometric, Riemann, and Lamé Equations

For the purposes of constructing explicit solutions to second-order linear homogeneous differential equations on the Riemann sphere the Kovacic algorithm partitions the subgroups ofSL (2,C) into four classes and initially determines which class contains the differential Galois group of the input equation. We prove in the case of the hypergeometric and Riemann equations that the relevant class can be determined directly from the coefficients by elementary calculation. We also treat the (non-algebraic form of the) Lamé equation, to which the Kovacic algorithm is not directly applicable. In that instance we combine the Kovacic results with ours to produce an algorithm for determining the class of the associated group. From the group-theoretic viewpoint the problem solved herein is the following: given arbitrary S,T SL(2,C), determine which class contains the group S, T generated by S and T.