Factorization formulas for higher depth determinants of the Laplacian on the n-sphere

TL;DR

This paper derives explicit factorization formulas for higher depth determinants of Laplacians on n-spheres, expressed through multiple gamma functions, advancing spectral analysis techniques.

Contribution

It provides the first explicit factorization formulas for higher depth determinants of Laplacians on n-spheres using multiple gamma functions.

Findings

Explicit formulas for higher depth determinants in terms of multiple gamma functions

Connections established between spectral zeta functions and special functions

Enhanced understanding of spectral invariants on n-spheres

Abstract

We explicitly give factorization formulas for higher depth determinants, which are defined via derivatives of the spectral zeta function at non-positive integer points, of Laplacians on the n-sphere in terms of the multiple gamma functions.