We define the higher order Riesz transforms and the Littlewood-Paley
$g$
-function associated to the differential operator
${{L}_{\lambda }}f(\theta )\,=\,-{f}''(\theta )-2\lambda \cot \theta {f}'(\theta )+{{\lambda }^{2}}f(\theta )$
. We prove that these operators are Calderón–Zygmund operators in the homogeneous type space
$((0,\,\pi ),\,{{(\sin t)}^{2\lambda }}dt)$
. Consequently,
${{L}^{p}}$
weighted,
${{H}^{1}}\,-\,{{L}^{1}}$
and
${{L}^{\infty }}\,-\,BMO$
inequalities are obtained.