Short version
What is the meaning of this sum notation? \begin{equation} \sum_{r = 0}^{\infty} \sum_{r_1 + \dots + r_p = r}^{} (\dots) \end{equation} I understand that $r = 0, 1, \dots, \infty$, but what about $r_1, \dots, r_p$? What values should I assign to $r_1, \dots, r_p$?
Long version (with context)
I am looking for a closed-loop expression of the expectation of the square root of a quadratic form $Q(X) = X' A X$ with $X \sim \mathcal{N}_p(\mu, \Sigma)$ and $A \succeq 0$.
The answer to this question mentions the book by Mathai & Provost, "Quadratic forms in Random Variables", that in Theorem 3.2b.5 reports the following equation for $\mathbb{E}[{Q^{h'}}]$ with $h' \in (0, 1)$. In my case $h' = h = 1/2$.
\begin{align} \mathbb{E}[{Q^{h'}}] &= \alpha^h \exp\left(-\sum_{j = 1}^{p} b_j^2 / 2 \right) \Biggl\{ \sum_{j = 1}^{p} b_j^2 \lambda_j \sum_{r = 0}^{\infty} \sum_{r_1 + \dots + r_p = r}^{} \frac{\left(\frac{b_1^2}{2}\right)^{r_1} \cdots \left(\frac{b_p^2}{2}\right)^{r_p}}{r_1 ! \cdots r_p !} \frac{\Gamma(\nu_1 + \dots + \nu_p + r - h)}{\Gamma(\nu_1 + \dots + \nu_p + r)} \nonumber \\ & F_D(h; \nu_1 + r_1, \dots, \nu_p + r_p; \nu_1 + \dots + \nu_p + r; 1 - 2 \alpha \lambda_1, \dots, 1 - 2 \alpha \lambda_p) \nonumber \\ & + \sum_{j = 1}^{p} \lambda_j \sum_{r = 0}^{\infty} \sum_{r_1 + \dots + r_p = r}^{} \frac{\left(\frac{b_1^2}{2}\right)^{r_1} \cdots \left(\frac{b_p^2}{2}\right)^{r_p}}{r_1 ! \cdots r_p !} \frac{\Gamma(\eta_1 + \dots + \eta_p + r - h)}{\Gamma(\eta_1 + \dots + \eta_p + r)} \nonumber \\ & F_D(h; \eta_1 + r_1, \dots, \eta_p + r_p; \eta_1 + \dots + \eta_p + r; 1 - 2 \alpha \lambda_1, \dots, 1 - 2 \alpha \lambda_p) \Biggr\} \end{align} where $\lambda_1, \dots, \lambda_p$ are the eigenvalues of $A \Sigma$ and $P$ is the orthogonal matrix of the eigenvectors of $A \Sigma$, that is \begin{align} [b_1,\, \dots,\, b_p]^T &= P^T \Sigma^{-1/2} \mu \\ \text{diag}\left( [\lambda_1,\, \dots,\, \lambda_p]^T \right) &= P^T \Sigma^{1/2} A \Sigma^{1/2} P \\ P P^T &= I \end{align} and \begin{align} \nu_i &= \begin{cases} \frac{1}{2} & \text{if } i \neq j \\ \frac{1}{2} + 2 & \text{if } i = j \end{cases} \\ \eta_i &= \begin{cases} \frac{1}{2} & \text{if } i \neq j \\ \frac{1}{2} + 1 & \text{if } i = j \end{cases} \end{align} $\alpha$ is an arbitrary parameter such that $\alpha > 0$ and $|1 - 2 \alpha \lambda_j| < 1$ for $j = 1, \dots, p$. Finally, $F_D$ is a Lauricella function.
Everything is clear to me except for the sums \begin{equation} \sum_{r = 0}^{\infty} \sum_{r_1 + \dots + r_p = r}^{} (\dots) \end{equation} that appear twice in the expression. I guess that the index $r$ is a sequence $r = 0, 1, \dots, \infty$. However, how am I supposed to pick the values of $r_1, \dots, r_p$? Can I pick them arbitrarily as a set of positive integers such that $r_1 + \dots + r_p = r$? Yet, the fact that there are two sums makes me think that I should pick multiple sets of $r_1, \dots, r_p$ for each $r$.