  The electron repulsion integrals over the Slater-type orbitals with
non-integer principal quantum numbers are considered. These integrals are
useful in both non-relativistic and relativistic calculations of many-electron
systems. They involve hyper-geometric functions. Due to the non-trivial
structure of infinite series that are used to define them the hyper-geometric
functions are practically difficult to compute. Convergence of their series are
strictly depends on the values of parameters. Computational issues such as
cancellation or round-off error emerge. Relationships free from
hyper$-$geometric functions for expectation values of Coulomb potential
$\left(r_{21}^{-1}\right)$ are derived. These relationships are new and show
that the complication coming from two-range nature of Laplace expansion for the
Coulomb potential is removed. These integrals also form an initial condition
for expectation values of a potential with arbitrary power. The electron
repulsion integrals are expressed by finite series of power functions. The
methodology given here for evaluation of electron repulsion integrals are
adapted to multi-center integrals.
