This is a quation about notation.
While doing quantum mechanics, I am faced with expanding the following expression: $$\Gamma_d^k |\psi \rangle $$
where $|\psi \rangle$ is an element of a Hilbert space (not important for this question). $\Gamma_d = \gamma_d \gamma_{-d}$ where $$ \gamma_d = \sum_{m=0}^{\mu} \left( a^\dagger_{m+d} a_{m} - b^\dagger_{m-d} b_{m} \right)$$
and $a_{p}^{(\dagger)}, b_{q}^{(\dagger)}$ are annihilation(creation) operators. Hence $\Gamma_d$ is a double sum over, say, $m,m'$ which, when expanded, would contain 4 terms for each value of $m,m'$. The number of terms is then $4(\mu + 1)^2$.
My question is: what is a good notation for "expanding" $\Gamma_d^k$? How would I write
$$\Gamma_d^k |\psi \rangle = \sum_{?}^{?} \cdots \sum_{??}^{??} T$$
so that $T$ is an expression for one of the $4^k(\mu + 1)^k$ terms?
Generally the best way to expand a product of sums is to put the indices into a vector.
$$\prod_{i\in I}\sum_{j\in J}x_{ij} = \sum_{j\in J^I}\prod_{i\in I}x_{ij_i}$$