Evaluation of spherical GJMS determinants

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.