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

Differential Geometry;Functional Analysis;Quantum Algebra