$y=y_N$ and $\bar k=k-y$. Write for : $$\begin{aligned}
        &
        \mu^k
        \sum_{\vec x'\in \mathbb{X}^{N}}
        \sum_{\vec y'\in \mathbb{Y}^{N}_k}
        \,\prod_{i=1}^N
        \mathbf{P}_{x_{i-1}-x_{i-1}',x_{i-1}-x_i-1}(x_i-x_i')
        \prod_{i=1}^{N}
        \varphi_{q,q/\mu,\nu}
        \left( \sum\nolimits_{j=0}^{i-1}(y_j'-y_j)\;\Big\vert\;y_i \right)
        \prod_{r=0}^{N}q^{y_r'(x_r'+r)}
        \\&\hspace{2pt}
        =
        \mu^{\bar k}
        \sum_{\vec x'\in \mathbb{X}^{N-1}}
        \sum_{\vec y'\in \mathbb{Y}^{N-1}_{\bar k}}
        \prod_{i=1}^{N-1}\mathbf{P}_{x_{i-1}-x_{i-1}',x_{i-1}-x_i-1}(x_i-x_i')
        \prod_{i=1}^{N-1}\varphi_{q,q/\mu,\nu}\left( \sum\nolimits_{j=0}^{i-1}(y_j'-y_j)\;\Big\vert\;y_i \right)
        \prod_{r=0}^{N-1}q^{y_r'(x_r'+r)}
        \\
        &\hspace{80pt}\times
        \mu^{y}
        \sum_{s=0}^{y}\varphi_{q,q/\mu,\nu}(s\mid y)\, q^{s(x_{N-1}'+N-1)}
        \sum_{L=0}^{\infty}
        q^{(y-s)(x_N-L+N)}\,\mathbf{P}_{x_{N-1}-x_{N-1}',x_{N-1}-x_N-1}(L).
    \end{aligned}$$ Here we used the notation $s=y_N-y_N'=\sum_{j=0}^{N-1}(y_j'-y_j)$ and $L=x_N-x_N'$. The factor $q^{s(x_{N-1}'+N-1)}$ appears on the last line because in the sum over $\vec{y}\,'\in\mathbb{Y}^{N-1}_{\bar k}$ in the second line the quantity $y_{N-1}'$ is different from $y_{N-1}'$ in the first line: the former not take into account $s$ Boson particles coming from $y_N$. Continuing the computation, we can now apply with $\ell=x_{N-1}-x_{N-1}'$, $g=x_{N-1}-x_N-1$. By the induction hypothesis: $$\begin{aligned}
        &=
        \mu^{\bar k}
        \sum_{\vec x'\in \mathbb{X}^{N-1}}
        \sum_{\vec y'\in \mathbb{Y}^{N-1}_{\bar k}}
        \prod_{i=1}^{N-1}\mathbf{P}_{x_{i-1}-x_{i-1}',x_{i-1}-x_i-1}(x_i-x_i')
        \prod_{i=1}^{N-1}\varphi_{q,q/\mu,\nu}\left( \sum\nolimits_{j=0}^{i-1}(y_j'-y_j)\;\Big\vert\;y_i \right)
        \prod_{r=0}^{N-1}q^{y_r'(x_r'+r)}
        \\
        &\hspace{82pt}\times
        q^{y(x_N+N)}
        \nu^y
        \sum_{t=0}^{y}
        q^{(x_{N-1}-x_N-1)t}\,
        \varphi_{q,q/\nu,\nu}(t\mid y)
        \\&
        =
        \nu^{k}\sum_{\vec y''\in \mathbb{Y}^{N-1}_{\bar k}}
        \sum_{t=0}^y
        \varphi_{q,q/\nu,\nu}(t\mid y)
        \prod_{i=1}^{N-1}\varphi_{q,q/\nu,\nu}
        \biggl( \sum\limits_{j=0}^{i-1}(y_j''-y_j)\;\Big\vert\;y_i \biggr)
        q^{t(x_{N-1}+N-1)+(y-t)(x_N+N)}
        \prod_{r=0}^{N-1}q^{y_r''(x_r+r)},
    \end{aligned}$$ where $t=y_N-y_N''=\sum_{j=0}^{N-1}(y_j''-y_j)$. We can rewrite the sum over $\vec y\,''\in \mathbb{Y}^{N-1}_{\bar k}$ and $t$ as a sum over $\vec y\,''\in \mathbb{Y}^N_k$, thus arriving at the right-hand side of . Throughout the whole computation, all infinite series converge for sufficiently small $\mu$ because all $y_i$, $y_i'$, $y_i''$ are 