On a compact Riemannian manifold $(M,g)$ without boundary it was shown (by R. Seeley) how to define the complex power of an elliptic classical pseudodifferential operator $A$ of positive order $m$: using the holomorphic functional calculus one has (under suitable conditions) $$ A^{-s} = \frac{i}{2 \pi} \int_\Gamma \,\lambda^{-s}(A - \lambda)^{-1} \,d\lambda $$ with $\Gamma$ a contour surrounding the spectrum of $A$. Essentially many properties of this operator can be deduced from the asymptotic expansion for the local resolvent symbol $$ \sigma_{(A - \lambda)^{-1}}(x,\xi,\lambda) \backsim \sum_{j \ge 0} b_{ - m - j}(x,\xi,\lambda) \quad (\text{as }(\xi,\lambda^{1/m}) \to \infty) $$ and there is an explicit recursive formula for the terms $b_{ - m - j}$, given by \begin{align} b_{-m} &= (a_m - \lambda)^{-1} \\ b_{ - m -j} &= - b_{-m}\sum_{\substack{l < j \\ l + k + |\alpha| = j}} \frac{1}{\alpha!}(\partial^\alpha_\xi b_{ - m - l})D^\alpha_x a_{m - k} \qquad (\text{for } j > 0) \end{align} where $a \backsim \sum_{i \ge 0} a_{m - i}$ is an asymptotic expansion for the local symbol $a$ of $A$.

I was wondering whether it is known if the same recursive formula for the resolvent symbols (the $b_{ - m - j}$) holds if one works on a noncompact manifold instead.