# A multiple integral related to the binomial theorem

Recently a multiple integral appeared in my research, and I wonder if there is some trick to compute this integral, and how it is connected to something well-known such as the binomial theorem, as symbolic integration in sagemath for several values of $N$ show (you will find the code following this link).

For $N\geq 1$, $\beta >0$, $s_N > 0$, let \begin{equation} T_N(s_N) = \int_{0}^{s_N} \int_{0}^{s_{N-1}} \cdots \int_{0}^{s_2} \int_{0}^{s_1} e^{\beta \left( -Ns_0 + \sum\limits_{i=1}^{N} s_i \right)} d s_0 d s_1 \ldots d s_{N-1}. \end{equation}

Prove (or disprove) that \begin{equation} T_N(s_N) = \frac{1}{\beta^{N}} \sum_{k=0}^{N} \frac{(-1)^{N-k}}{(N-k)!k!}e^{\beta k s_N} = \frac{(e^{\beta k s_N} - 1)^N}{N!\beta^{N}}. \end{equation}

Note 1: I tried to prove it by induction but I don't know how to connect $T_N$ with $T_{N+1}$. Note that \begin{equation} T_{N+1}(s_{N+1}) = e^{\beta s_{N+1}} \int_{0}^{s_{N+1}}\left[ \int_{0}^{s_{N}} \int_{0}^{s_{N-1}} \cdots \int_{0}^{s_2} \int_{0}^{s_1} e^{-\beta s_0} e^{\beta \left( -Ns_0 + \sum\limits_{i=1}^{N} s_i \right)} d s_0 d s_1 \ldots d s_{N-1} \right] d s_N \end{equation}

Simply observe that \begin{equation} \int_{0}^{s_N} \int_{0}^{s_{N-1}} \cdots \int_{0}^{s_2} \int_{0}^{s_1} e^{\beta \left( -Ns_0 + \sum\limits_{i=1}^{N} s_i \right)} d s_0 d s_1 \ldots d s_{N-1} = \int_{0\leq s_0\leq s_1\leq \ldots \leq s_n}e^{\beta \left( -Ns_0 + \sum\limits_{i=1}^{N} s_i \right)} d s_0 d s_1 \ldots d s_{N-1}\,, \end{equation} that is, we integrate on $(s_0,\ldots,s_{N-1})$ on the region where $0\leq s_0\leq s_1\leq \ldots \leq s_N$. Let us observe that the integrand is symmetric on $s_1,\ldots,s_{N-1}$ thus it follows that leaving out the order condition $s_1\leq s_2\leq \ldots \leq s_{N-1}$ we get $$T_N = \frac{1}{(N-1)!} \int_{0}^{s_N}\left(\int_{s_0}^{s_N} \int_{s_0}^{s_N} \cdots \int_{s_0}^{s_N} e^{\beta \left( -Ns_0 + \sum\limits_{i=1}^{N} s_i \right)} d s_1 \ldots d s_{N-1} \right)d s_0$$ (why? ;) we are considering that all possible permutations give us the same)
Now this reduces to $$T_N = \frac{\exp(\beta s_N)}{(N-1)!} \int_0^{s_N} \exp(-\beta N s_0)\,\left(\int_{s_0}^{s_N} \exp(\beta \,s) ds\right)^{N-1} ds_0$$
Here compute the integral inside the power (to the $N-1$) and use the Binomial Theorem.