Quadratic Transformations of the Sixth Painleve' Equation with Application to Algebraic Solutions

In 1991, one of the authors showed the existence of quadratic transformations between the Painleve' VI equations with local monodromy differences $(1/2,a,b,\pm 1/2)$ and $(a,a,b,b)$. In the present paper we give concise forms of these transformations. %, up to fractional-linear transformations. They are related to the %better known quadratic transformations obtained by Manin and Ramani-Grammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic-pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painleve' VI functions.