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

TL;DR

This paper extends the Connes-Moscovici functional relations to higher-dimensional noncommutative tori, introducing transformations that facilitate systematic analysis and enable computer-free verification of complex hypergeometric relations.

Contribution

It introduces a systematic approach with transformations for analyzing spectral functions, extending functional relations to arbitrary dimensions, and enabling verification without computational aid.

Findings

Extended Connes-Moscovici relations to higher dimensions

Developed transformations for spectral function analysis

Achieved computer-free verification of hypergeometric relations

Abstract

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.