I am searching for a solution to the following integral:
\begin{align} \mathbf{L}^{k}(x)\colon& = \int_{0}^{x}\int_{0}^{v_{k}}\int_{0}^{t_{k}}\int_{0}^{v_{k-1}}\\& \dots \int_{0}^{t_{2}}\int_{0}^{v_{1}}\mathrm{e}^{\sum_{i=1}^{k}\frac{v_{i}-t_{i}}{\lambda}}\quad \mathrm{d}t_{1}\mathrm{d}v_{1}\dots \mathrm{d}{t_{k-1}}\mathrm{d}{v_{k-1}}\mathrm{d}{t_{k}}\mathrm{d}{v_{k}} \end{align}
where $\lambda>0$. I note this integral is related to the question here. However, this time, the function $f$ that we are integrating has two arguments and we alternate the repeated integration over these two arguments. The function we are integrating is also not symmetric in the two arguments.
Letting $\lambda =1$ to start with something simple, based on the solution for the Dyson series, my first attempt was to try and prove the following:
\begin{multline} \mathbf{L}^{k}(x)= (1/k!)\left(\int_{0}^{x}\int_{0}^{v} e^{v-t}\,\,\mathrm{d}t\mathrm{d}v \right)^{k} \end{multline}
However, this is not correct, as one can check via induction.
EDIT:
Thanks to the comments, I now realise this is a generalisation of a Dyson series. Compared to the Dyson series, the sign of the argument now alternates. The Dyson series has received plenty of attention, for example here and here
In particular, we now seek a solution to:
\begin{align} \mathbf{L}^{k}(x)\colon& = \int_{v_{0}}^{x}\int_{v_{0}}^{s_{k}} \dots \int_{v_{0}}^{s_{3}}\int_{v_{0}}^{s_{2}}\Pi_{i=1}^{k}f((-1)^{i}s_{i})\quad \mathrm{d}s_{1}\mathrm{d}s_{2}\dots \mathrm{d}{s_{k-1}}\mathrm{d}{s_{k}} \end{align}
The trouble with the methods already highlighted in the links above is that the function $$f_{n}\colon s_{1}, s_{2}, \dots s_{n}\mapsto f(-s_{1})\times f(s_{2})\dots \times f(-1^{n}s_{n})$$ is not symmetric.
This matters because we cannot find a "hypercube" that we can integrate over in a way that all the integral bounds are identical (we then divide the integral over the hypercube into simplicies over which the integral is identical).
To take the example of $k=2$, consider integrating $f(s_{2}-s_{1})$ over the region A in the picture below:
Following the exposition of the Dyson series here 3, we cannot integrate $f(s_{2}-s_{1})$ over the region A+B and evaluate $(1/2)\int_{0}^{x}\int_{0}^{x}f(s_{2}-s_{1})ds1 ds2$ since the integral over the two regions is not symmetric. A is however identical to integrating over the region C, however, I cannot figure out how to use this fact to give a repeated integral whose bounds are identical.
If anyone has any suggestions or solutions, I would really appreciate and be grateful!
Edit 2:*
We can also frame this question as a series of nested double integrals. This is unanswered here.