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We give a new proof of the rearrangement lemma that works for all dimensionsand all heat coefficients in the study of modular geometry on noncommutativetori. The building blocks of the spectral functions are landed in ahypergeometric family knowns as Lauricella functions of type $D$. Weinvestigate the differential and recursive relations among the family andobtain a full reduction to Gauss hypergeometric functions. As for applications,we give phase one demonstration on how the hypergeometric features lead to newsimplifications in computations involved in the modular geometry.

Hypergeometric function and Modular Curvature I. --Hypergeometric functions in Heat Coefficients