  Given two holomorphic functions $f$ and $g$ defined in two respective germs
of complex analytic manifolds $(X,x)$ and $(Y,y)$, we know thanks to M. Saito
that, as long as one of them is Euler homogeneous, the reduced (or microlocal)
Bernstein-Sato polynomial of the Thom-Sebastiani sum $f+g$ can be expressed in
terms of those of $f$ and $g$. In this note we give a purely algebraic proof of
a similar relation between the whole functional equations that can be applied
to any setting (not necessarily analytic) in which Bernstein-Sato polynomials
can be defined.
