  Complex-valued functions defined on a finite interval $[a,b]$ generalizing
power functions of the type $(x-x_0)^n$ for $n\geq 0$ are studied. These
functions called $\Phi$-generalized powers, $\Phi$ being a given nonzero
complex-valued function on the interval, were considered to contruct a general
solution representation of the Sturm-Liouville equation in terms of the
spectral parameter \cite{kravchenko2008, kravporter2010}. The
$\Phi$-generalized powers can be considered as a natural basis functions for
the one-dimensional supersymmetric quantum mechanics systems taking
$\Phi=\psi_0^2$, where the function $\psi_0(x)$ is the ground state wave
function of one of the supersymmetric scalar Hamiltonians. Several properties
are obtained such as $\Phi$-symmetric conjugate and antisymmetry of the
$\Phi$-generalized powers, a supersymmetric binomial identity for these
functions, a supersymmetric Pythagorean elliptic (hyperbolic) identity
involving four $\Phi$-trigonometric ($\Phi$-hyperbolic) functions as well as a
supersymmetric Taylor series expressed in terms of the $\Phi$-derivatives. We
show that the first $n$ $\Phi$-generalized powers are a fundamental set of
solutions associated with a nonconstant coefficients homogeneous linear
ordinary differential equations of order $n+1$. Finally, we present a general
solution representation of the stationary Schr\"odinger equation in terms of
geometric series where the Volterra compositions of the first type is
considered.
