the $\tilde c'_k$ vanish.

As explained e.g. in [@GS2] this yields the expansions: $$\align
\Tr(Ae^{-tP_{1,\operatorname{D}}})\sim
\sum_{j\ge 0}
c_{j} t
^{\frac{j-n-\nu  }{2}}
+\sum_{k\ge 0}(-  c'_k\log t +  c''_k)t
^{\frac k2},\tag1.11\\
 \Gamma (s)\Tr
(AP_{1,\operatorname{D}}^{-s})\sim \sum_{j\ge 0}\frac{
c_j }
{s+\frac{j-n-\nu  }{2}}
+\sum_{k\ge 0}\Bigl(\frac{  c'_k}{(s+\frac k2)^2} +\frac{
c''_k}
{s+\frac k2}\Bigr) .\tag1.12
\endalign$$ In (1.10), $\lambda \to\infty $ in a sector of $\Bbb C$; in (1.11), $t\to 0+$; (1.12) describes the pole structure of the meromorphic extension of $\Gamma (s)\Tr
(AP_{1,\operatorname{D}}^{-s})$ to $\Bbb C$. The coefficients $c_j,  c_k', c''_k$ are proportional to the coefficients $\tilde c_j, \tilde c_k', \tilde
c''_k$ in (1.10) by universal constants; in particular,$$c'_0=\tilde c'_0,\quad c''_0=\tilde c''_0, \quad c_{\nu +n}=\tilde
c_{\nu +n},\tag1.13$$ where $c_{\nu +n}$ and $\tilde c_{\nu +n}$ are taken as 0 when $\nu
<-n$ or $\nu \notin\Bbb Z$. The coefficients $c_j$ and $c_k'$ are locally determined, whereas the $c''_k$ are global.

Relation (1.12) implies that the generalized zeta function $\zeta
(A,P_{1,\operatorname{D}},s)=\Tr (AP^{-s}_{1,\operatorname{D}})$ extends meromorphically with $$\gathered
\zeta
(A,P_{1,\operatorname{D}},s)=C_{-1}(A,P_{1,\operatorname{D}})s^{-1}
+C_0(A,P_{1,\operatorname{D}})+O(s) \text{ for }s\to 0,\\
C_{-1}(A,P_{1,\operatorname{D}})=\tilde c'_0,\quad
C_0(A,P_{1,\operatorname{D}})=\tilde c_{\nu +n}+\tilde c''_0.
\endgathered\tag1.14$$ In [@FGLS], an analogue of Wodzicki’s noncommutative residue was introduced by Fedosov, Golse, Leichtnam, and Schrohe for operators in Boutet de Monvel’s calculus. In [@GSc] we were able to show 