A note on the functional determinant of higher-derivative scalar fields on sphere products

It is shown that the functional determinant ($\sim$ effective action) for a scalar field propagating on the mixed signature product of unit spheres, S$^q\times$S$^p$, according to the GJMS operator, depends, if $d$ is odd, only on $d=p+q$ and on whether $p$ is even or odd. In the first case the effective action equals twice the standard quantity on S$^d$ and vanishes in the second.