cf. Grubb and Seeley [@GS1]. Here each $\tilde c_j$ (and each $\tilde c_k'$) comes from a specific homogeneous term in the symbol of $A(P-\lambda)^{-N}$, whereas the $\tilde c''_k$ depend on the full symbol. So the coefficients $\tilde c_j$ and $\tilde c_k'$ depend each on a finite set of homogeneous terms in the symbols of $A$ and $P$; we call such coefficients ‘locally determined’ (or ‘local’), while the $\tilde
c''_k$ are called ‘global’. When $\nu \notin
\Bbb Z$, the $\tilde c'_k$ vanish. When $\nu \in\Bbb Z$ and $(j-n-\nu
)/m$ is an integer $k\ge 0$, both $\tilde c_j$ and $\tilde c''_k$ contribute to the power $(-\lambda )^{-k-N}$; their sum is independent of the choice of local coordinates, whereas the splitting in $\tilde c_j$ and $\tilde c''_k$ depends in a well-defined way on the symbol structure in local coordinates (see or [@G4; @Th.; @1.3]).

In (1.1), $m\cdot \tilde c_0'= \operatorname{res}A$, depending solely on $A$, cf. [@W], [@Gu]. Moreover, cf. [@KV], [@G4], $$\operatorname{TR} A= \tilde c''_0\tag1.2$$ in the following four cases:

“(1)” $\nu <-n$,

“(2)” $\nu \notin\Bbb Z$,

“(3)” $\nu \in\Bbb Z$, $A$ is even-even and $n$ is odd,

“(4)” $\nu \in\Bbb Z$, $A$ is even-odd and $n$ is even; in the cases (3) 