  In the paper, the well-known quantum mechanical problem of a spin 1/2
particle in external Coulomb potential, reduced to a system of two first-order
differential equations, is studied from the point of view of possible
applications of the Heun function theory to treat this system. It is shown that
in addition to the standard way to solve the problem in terms of the confluent
hypergeometric functions (proposed in 1928 by G. Darvin and W. Gordon), there
are possible several other possibilities which rely on applying the confluent
Heun functions. Namely, in the paper there are elaborated two combined
possibilities to construct solutions: the first applies when one equation of
the pair of relevant functions is expressed trough hypergeometric functions,
and another constructed in terms of confluent Heun functions. In this respect,
certain relations between the two classes of functions are established. It is
shown that both functions of the system may be expressed in terms of confluent
Heun functions. All the ways to study this problem lead us to a single energy
spectrum, which indicates their correctness.
