Page "Painlevé transcendents" Paragraph 5

found that forty-four of the fifty equations are reducible in the sense that they can be solved in terms of previously known functions, leaving just six equations requiring the introduction of new special functions to solve them.

( There were some computational errors in his work, which were fixed by B. Gambier and R.

Fuchs.

) It was a controversial open problem for many years to show that these six equations really were irreducible for generic values of the parameters ( they are sometimes reducible for special parameter values ; see below ), but this was finally proved by and.