In the paper titled "Functional Analysis of Nonlinear Circuits: A Generating Power Series Approach" by M. Lamnabhi, Proposition 3 (located at the beginning of page 382) provides the Laplace-Borel transform of the response associated with a Dirac input. However, the paper does not include any reference or derivation for this result. I am interested in understanding the process of obtaining this result since it differs slightly from the deterministic counterparts. I have reviewed the literature of other closely affiliated authors, such as M. Fleiss and F. Lamnabhi-Lagarrigue, but I have been unable to find any information regarding this derivation. My suspicion is that the result may be contained within M. Lamnabhi's thesis titled "Analyse des Systèmes Non Linéaires par les Méthodes de Développements Fonctionnels," but I have been unable to locate an online version of this thesis.
The paper introduces a generating power series given by:
\begin{equation} g_n = (1 + a_0x_0)^{-1}x_{i_1}(1+a_1x_0)^{-1} \dots (1+a_{n-1}x_0)^{-1}x_{i_n}(1+a_nx_0)^{-1} = (1+a_0x_0)^{-1}x_{i_1}g_{n-1} \end{equation} where $k$ represents some constant. Furthermore, the response to a Dirac input is defined over its length as follows:
\begin{equation} \text{IRF}(g_n) = \begin{cases} (1+a_0x_0)^{-1}x_0 \cdot \text{IRF}(g_{n-1}), & i_1=0 \\ \frac{1}{n!} (1 + a_0 x_0)^{-1}, & i_1 = 1, \text{ and } i_2 = \ldots = i_n = 1\\ 0, & i_1=1 \text{ and }\exists p,\; p>1,\; i_p \neq 1 \end{cases} \end{equation}
I have attempted to derive the Laplace-Borel transforms using the standard approach by considering a general analytic function represented as a power series:
\begin{equation} h(t) = \sum_{i \geq 0} h_i \frac{t^i}{i!} = \sum_{i \geq 0} h_i \int^t_0 \mathrm{d}\tau_n \int^{\tau_n}_0 \mathrm{d}\tau_{n-1} \dots \int_0^{\tau_2} \mathrm{d}\tau_1 \end{equation}
This leads to the Laplace-Borel transformed analogue:
\begin{equation} g_h = LB[h(t)] = \sum_{i \geq 0} h_i x_0^i \end{equation}
EDIT1: After some subsequent attempts, I think it's more informative to approach the problem in terms of repeated integration and the analogue form for the Volterra kernels. So consider the same generating series as above, its time domain analogue is represented by the repeated integral:
\begin{equation} \int_0^t \int_0^{\tau_n}\dots \int_0^{\tau_2} e^{a_0(t-\tau_n)}e^{a_1(\tau_n-\tau_{n-1})} \dots e^{a_{n-1}(\tau_2-\tau_1)}e^{a_n\tau_1}u_{i_n}(\tau_1) \dots u_{i_2}(\tau_{n-1})u_{i_1}(\tau_n)\mathrm{d}\tau_1\dots\mathrm{d}\tau_n \end{equation}
Where $u_0(t)=1$ and $u_1(t) = \delta(t)$.
I believe this form of the equation can be rearranged into the form (with the corresponding terms in the transformed domain shown in the underbraces):
\begin{equation} \underbrace{\int_0^t e^{a_0(t-\tau_n)}}_{(1-a_0x_0)^{-1}} \underbrace{\int_0^{\tau_n} u_{i_1}(\tau_n)e^{a_1(\tau_n-\tau_{n-1})}}_ {x_{i_1}(1-a_1x_0)^{-1}} \dots \underbrace{\int_0^{\tau_3} u_{i_{n-1}}(\tau_2)e^{a_{n-1}(\tau_2-\tau_1)}}_{x_{i_{n-1}}(1-a_{n-1}x_0)^{-1}} \underbrace{\int_0^{\tau_2}u_{i_n}(\tau_1) e^{a_n\tau_1}}_{x_{i_n}(1-a_nx_0)^{-1}} \mathrm{d}\tau_1 \dots \mathrm{d}\tau_n \end{equation}
Consider the case where $i_1=0$. As stated above this will result in $u_0(t)=1$.
\begin{equation} \underbrace{\int_0^t e^{a_0(t-\tau_n)}}_{(1-a_0x_0)^{-1}} \underbrace{\int_0^{\tau_n}e^{a_1(\tau_n-\tau_{n-1})}}_ {x_0(1-a_1x_0)^{-1}} \dots \underbrace{\int_0^{\tau_3} u_{i_{n-1}}(\tau_2)e^{a_{n-1}(\tau_2-\tau_1)}}_{x_{i_{n-1}}(1-a_{n-1}x_0)^{-1}} \underbrace{\int_0^{\tau_2}u_{i_n}(\tau_1) e^{a_n\tau_1}}_{x_{i_n}(1-a_nx_0)^{-1}} \mathrm{d}\tau_1 \dots \mathrm{d}\tau_n \end{equation}
which essentially gives the first case shown in the paper when defining recursively over the length.
However, when you consider the case when $i_1=1$ we have $u_1(t) = \delta(t)$, resulting in the form: \begin{equation} \underbrace{\int_0^t e^{a_0(t-\tau_n)}}_{(1-a_0x_0)^{-1}} \underbrace{\int_0^{\tau_n}\delta(\tau_n)e^{a_1(\tau_n-\tau_{n-1})}}_{x_1(1-a_1x_0)^{-1}} \dots \underbrace{\int_0^{\tau_3} u_{i_{n-1}}(\tau_2)e^{a_{n-1}(\tau_2-\tau_1)}}_{x_{i_{n-1}}(1-a_{n-1}x_0)^{-1}} \underbrace{\int_0^{\tau_2}u_{i_n}(\tau_1) e^{a_n\tau_1}}_{x_{i_n}(1-a_nx_0)^{-1}} \mathrm{d}\tau_1 \dots \mathrm{d}\tau_n \end{equation}
I do not understand how they obtain the result where if there exists a $u_0(t)$ term later in the sequence the whole series collapses to zero. Please could someone assist in this case, as my knowledge of Dirac-deltas is limited?