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In his book Aspects of Quantum Field Theory in Curved Spacetime Stephen Fulling calculates the coincidence limit $[a_1]$ and gives an idea of how $[a_n]$ with $2 ≤ n$ can be found recursively.

Since the calculation of $[a_2]$ is much more laborious than the calculation of $[a_1]$, I hope that it can be found in the literature. Some references would be welcome.

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    $\begingroup$ I’ve deleted my answer. I hope that someone else can help you more than I did. $\endgroup$
    – Ghoster
    Commented Mar 16, 2023 at 21:26
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    $\begingroup$ @Filippo there is a book by SA Fulling where the procedure you are interested in is explained in some details. It is entitled Aspects of QFT in curved spacetime. $\endgroup$
    – Valter Moretti
    Commented Mar 16, 2023 at 21:37
  • $\begingroup$ By the way, I suspect that the root of our argument over my now-deleted answer is that your paper (which I could not see) factors out $\mathscr D^{1/2}$ whereas my answer factored out $\Delta^{1/2}$. If you had provided sufficient information in your question, I would not have wasted an hour writing an answer that did not match what your paper does. $\endgroup$
    – Ghoster
    Commented Mar 16, 2023 at 21:38
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    $\begingroup$ cambridge.org/core/books/… $\endgroup$
    – Valter Moretti
    Commented Mar 16, 2023 at 21:43
  • $\begingroup$ @Ghoster I am sorry. But both my paper and your paper say that $a_0$ is defined by $\langle\mathrm d\sigma,\nabla a_0\rangle=0$ and $[a_0]=1$ and I don't think that $a_0\equiv 1$ is a solution. Here is another argument, that you may not like: In the language of mathematicians, we assume that we are given a vector bundle $E\to M$ and a generalized Laplacian $F\colon\Gamma(M,E)\to\Gamma(M,E)$ (there exist $\nabla$ and $P$ s.t. $F=\nabla^*\nabla+P$). Then it doesn't make sense to assume that $x\mapsto a_0(x,x')\in L(E_{x'},E_x)$ (defined on a neighborhood of $x'$) is constant. $\endgroup$
    – Filippo
    Commented Mar 16, 2023 at 22:04

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