  Our understanding of local index formula in noncommutative geometry is
stalled for a while because we do not have more than one explicit computation,
namely that of Connes for quantum SU(2) and do not understand the meaning of
the various multilinear functionals involved in the formula. In such a
situation further progress in understanding necessitates more explicit
computations and here we execute the second explicit computation for the
quantum double suspension, a construction inspired by the Toeplitz extension.
More specifically we compute local index formula for the quantum double
suspensions of $C(S^2)$ and the noncommutative $2$-torus.
