  As the second part of the sequel, we investigate the variation of
rearrangement operators (more precisely, the spectral functions behind) arising
in the study of modular geometry on noncommutative (two) tori. We initiate a
systematic approach by introducing transformations corresponding to basic
operations in calculus, like differentiation and integration by parts. As for
applications, we extend, in a uniform way, the Connes-Moscovici's functional
relations on noncommutative two tori attached to the variation of second heat
coefficients to noncommutative tori of arbitrary dimension. Moreover, those
transformations lead to more internal relations among the hypergeometric family
obtained in part I of the sequel, which allows us to obtain, the first time, a
computer-aid free verification of those Connes-Moscovici type functional
relations.
