In the present article we introduce and study a novel type of solutions to the general Heun's equation. Our approach is based on the symmetric form of the Heun's differential equation yielded by development of the Papperitz-Klein symmetric form of the Fuchsian equations with an arbitrary number N≥4 of regular singular points. We derive the symmetry group of these equations which turns to be a proper extension of the Mobius group. We also introduce and study new series solutions of the proposed in the present paper symmetric form of the general Heun's differential equation (N=4) which treats simultaneously and on an equal footing all singular points.
International Frontier Science Letters (Volume 7)
P. Fiziev "Novel Representation of the General Fuchsian and Heun Equations and their Solutions", International Frontier Science Letters, Vol. 7, pp. 11-24, 2016