Connes' calculus for The Quantum double suspension

Given a spectral triple $(\mathcal{A},\mathcal{H},D)$ Connes associated a canonical differential graded algebra ${\mathrm{\Omega }}_D^{\bullet }(\mathcal{A})$. However, so far this has been computed for very few special cases. We identify suitable hypotheses on a spectral triple that helps one to compute the associated Connes’ calculus for its quantum double suspension. This allows one to compute ${\mathrm{\Omega }}_D^{\bullet }$ for spectral triples obtained by iterated quatum double suspension of the spectral triple associated with a first order differential operator on a compact smooth manifold. This gives the first systematic computation of Connes’ calculus for a large family of spectral triples.