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As the second part of the sequel, we investigate the variation ofrearrangement operators (more precisely, the spectral functions behind) arisingin the study of modular geometry on noncommutative (two) tori. We initiate asystematic approach by introducing transformations corresponding to basicoperations in calculus, like differentiation and integration by parts. As forapplications, we extend, in a uniform way, the Connes-Moscovici's functionalrelations on noncommutative two tori attached to the variation of second heatcoefficients to noncommutative tori of arbitrary dimension. Moreover, thosetransformations lead to more internal relations among the hypergeometric familyobtained in part I of the sequel, which allows us to obtain, the first time, acomputer-aid free verification of those Connes-Moscovici type functionalrelations.

Hypergeometric function and Modular Curvature II. Connes-Moscovici functional relation after Lesch's work