  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.
