Other reductions of to a , with at least one free parameter, exist iff the pair takes one of a finite number of values, where . Below are three such reductions with three and two parameters. They are analogous to quadratic and cubic hypergeometric transformations (§§15.8(iii)–15.8(v)).
equation (31.2.1) becomes Lamé’s equation with independent variable ; compare (29.2.1) and (31.2.8). The solutions (31.3.1) and (31.3.5) transform into even and odd solutions of Lamé’s equation, respectively. Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities , , and , where and are related to as in §19.2(ii).