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By introducing a new parameter as a second associated index for
special functions, we construct the three-dimensional differential
generators of $gl(2,\emph{c})$ Lie algebra together with the
corresponding contracted form $h_{4}$. Non-Casimir quadratic as
well as the Casimir of $gl(2,\emph{c})$ $(and h_{4})$ generators
can be considered as quantum solvable models on group manifold
$SL(2,\emph{c})$. Then, by appropriate parametrization of group
manifold $SL(2,\emph{c})$ and eliminating one of the coordinates,
we obtain quantum solvable Hamiltonians on homogeneous manifold
$SL(2,\emph{c})\div GL(1,\emph{c})$ with the metric described by
master function. We show that two-dimensional Hamiltonian on
$SL(2,\emph{c})\div GL(1,\emph{c})$ derived from the reduction of
Casimir operator $so(4,\emph{c})$ Lie algebra as a
three-dimensional Hamiltonian on group manifold $SL(2,\emph{c})$,
possesses the degeneracy $SL(2,\emph{c})$ group and, also, the
shape invariance property, where both have para-supersymmetry
representations of arbitrary order.
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