Evaluation of spherical GJMS determinants

TL;DR

This paper derives explicit formulas for the determinants of scalar GJMS operators on odd-dimensional spheres, linking them to integrals, sums, and special functions like the Riemann zeta and Bernoulli numbers.

Contribution

It provides a new integral and sum formula for GJMS determinants, connecting them to classical special functions and factorial coefficients, enhancing computational and theoretical understanding.

Findings

Explicit integral expression for GJMS determinants

Sum formula relating GJMS determinants to conformal Laplacian determinants

Connection to Riemann zeta function and Bernoulli numbers

Abstract

An expression in the form of an easily computed integral is given for the determinant of the scalar GJMS operator on an odd--dimensional sphere. Manipulation yields a sum formula for the logdet in terms of the logdets of the ordinary conformal Laplacian for other dimensions. This is formalised and expanded by an analytical treatment of the integral which produces an explicit combinatorial expression directly in terms of the Riemann zeta function, and $\log2$. An incidental byproduct is a (known) expression for the central factorial coefficients in terms of higher Bernoulli numbers.