Let $n \ge 1$ be an integer. Now let $0 \le t_0 \le t$ and $\beta \neq 1$ be real numbers. Now, let $\vec{p} := (p_0,p_1,\cdots,p_n)$ be strictly positive integers. Also let $(x)_{(n)} := x(x-1)\cdot \dots \cdot (x-n+1)$ be the Pochhammer symbol and let $s(p,k)$ by the Stirling number of the first kind. By generalizing the approach from Yet another multivariable integral over a simplex I have shown that the following identity holds: \begin{eqnarray} &&{\mathcal I}^{(t,t_0)}_n\left(\beta\right) := \int\limits_{t \le \xi_{n-1} \le \cdots \le \xi_0 \le t_0} \prod\limits_{j=0}^n (\xi_{j-1}-\xi_j)_{(p_j)} \cdot \prod\limits_{j=0}^{n-1} \frac{d\xi_j}{\xi_j^{2 \beta p_{j+1}}} = (-1)^n\sum\limits_{k_0=1}^{p_0} \dots \sum\limits_{k_n=1}^{p_n}\\ && t_0^{K_0} \sum\limits_{m=0}^n \sum\limits_{l=0}^{k_{n-m}} t^{(K_n-K_{n-m})-2 \beta(P_n-P_{n-m}) + m + l} t_0^{(K_{n-m}-K_0)-2 \beta (P_{n-m}-P_0)+n-m-l} \cdot \\ && \prod\limits_{j=0}^n s(p_j,k_j) \cdot\frac{\binom{k_{n-m}}{l} (-1)^{l+k_{n-m}+K_n}}{ \prod\limits_{\stackrel{j=0}{j\neq n-m}}^n (k_j+1) \binom{l+j-(n-m)-(K_{n-m}-K_j)+2 \beta (P_{n-m}-P_j)}{k_j+1}} \end{eqnarray} Here $P_j := p_0+ \dots + p_j$ and $K_j := k_0 + \dots + k_j$ .
Note 1: In the sum on the right hand side let us take only the terms where $k_j=p_j$ for $j=0,\dots,n$. Since $s(p_j,p_j)=1$ we note that the right hand side equals to the quantity $J^{(t,t_0)}_{\vec{p},n}$ in Yet another multivariable integral over a simplex .
Note 2: Let $p_0=p_1= \dots = p_n = 1$ meaning that $\sum\limits_{j=0}^n p_j = n$. Then of course $k_0=\dots = k_n=1$ and since $s(p_j,k_j) = 1$ we note that the right hand side is equal to the quantity $I_n^{(t,t_0)}$ from Multivariable integral over a simplex as it should be. That quantity has a neat closed form.
Note 3: The number of terms on the right hand side reads $1/2 \left(\prod\limits_{j=0}^n p_j \right) \left[3(n+1)+ \sum\limits_{j=0}^n p_j\right]$.
Now my question is what happens if we sum up the right hand side over all possible sequences $\vec{p}$ subject to $\sum\limits_{j=0}^n p_j = n + q$ where $q$ is some strictly positive integer. Will the result be also given by some neat closed form expression ?
