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Using the associated hypergeometric differential
equation, we solve analytically the bound states corresponding to
a hierarchy of the radial potential $-v_{0} e^{-\delta r}
/(1-e^{-\delta r}) + c\,e^{-\delta r} / (1-e^{-\delta r})^{2}$ as a
generalisation of the Hulth\'en potential. Then, an analytic
solution corresponding to a special case for which the parameter
$c$ is expected to be in terms of $l(l+1)$ is also derived.
Meanwhile without introducing a superpotential and in the framework of
supersymmetric quantum mechanics, it is shown that these bound
states can be calculated by two different algebraic methods. Based
on these two approaches, it is noticed that the bound states
realise an extended supersymmetry structure.
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