Question

If $p$ is a polynomial in $n$ real variables that is positive on $[0,1]^n$, the integral $\int_{[0,1]^n} p(\mathbf{x})^s d\mathbf{x}$ converges for $s$ a complex number with positive real part. In order to continue this function of $s$ to a meromorphic complex function elsewhere, you can employ the Bernstein-Sato theorem: there is a polynomial $b(s)$ and a (noncommutative) polynomial $D(x_i, \frac{\partial}{\partial x_i}, s)$, such that $b(s)p(\mathbf{x})^s = D\cdot p(\mathbf{x})^{s+1}$. This lets you expand the integral in terms of $p(\mathbf{x})^{s+1}$, but the new integral has extra terms attached. In order to make an inductive proof of continuation work, you need to prove that a more general class of integrals can be continued, namely products of terms of the form $p(\mathbf{x})^s$ with polynomials.