Suppose to have a chain (of size $L$) with bosons, and $\hat{a}\_i^\dagger$,$\hat{a}\_i$ are the associated creation and annihilation operators at site $i$. A Fock state can be written as:
\begin{equation}
| n\_1 \dots n\_L \rangle = \prod\_{i} \frac{1}{\sqrt{n\_i!}} \left( \hat{a}\_i^\dagger \right)^{n\_i} |\rangle
\end{equation}
where $|\rangle$ is the empty state. Now we define a new set of bosons:
\begin{align}
\hat{b}\_{k} &= \frac{1}{\sqrt{L}} \sum\_{j} e^{-ikj} \hat{a}\_j &
\hat{b}\_{k}^\dagger &= \frac{1}{\sqrt{L}} \sum\_{j} e^{ikj} \hat{a}\_j^\dagger
\end{align}
where $k$ are such that some boundary condition is fulfilled. Now a Fock state can be written as:
\begin{equation}
| \dots \tilde{n}\_k \dots \rangle = \prod\_{k} \frac{1}{\sqrt{\tilde{n}\_k!}}
\left( \hat{b}\_k^\dagger \right)^{\tilde{n}\_k} |\rangle
\end{equation}
The question is: is there a simple formula to express or compute the scalar product
$\langle \dots \tilde{n}\_k \dots | n\_1 \dots n\_L \rangle$?