  The Heun's equation with its four regular singularities emerges in many
applications in science. Despite the growing interest of the scientific
community, the literature has many gaps in conceptual mathematical aspects of
this equation. Moreover, the translation of the mathematical language for
non-mathematicians in making contemporary ideas applied in physical research is
not also well developed. In this paper, Maier's Heun-to-hypergeometric
reduction cases are studied in detail for four problems in quantum mechanics:
(1) The Schrodinger's equation for the Coulomb problem on a 3-sphere, (2) the
s-wave bound state equation in the problem of the attractive inverse square
potential, (3) the equation for the limit density function for the
discrete-time quantum walk, and (4) charged particle under magnetic field and
Coulomb force. All these problems give rise to the Heun's equation in their
mathematical formulation. A Sage code is given in the appendix for the
calculation of some Heun identities.
