Integral transformation of Heun's equation and some applications
It is known that the Fuchsian differential equation which produces the sixth Painlev� e equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parame- ters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency of a singularity. For the elliptical representation of Heun's equation, correspondence with respect to monodromy implies isospec- tral symmetry. We apply the symmetry tonite-gap potentials and express the monodromy of Heun's equation with parameters which have not yet been studied.