probability $\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)$ due to the push of the $i-1$-st particle. After that, the $i$-th particle makes an extra move $x_{i}(t)-p \to x_{i}(t)-p-m$ (where $m+p=L$) with probability $\psi_{q, \nu \mu^{-1} q^{p}, \nu q^{g}, \nu^{2} q^{g+p}}(m) \geq 0$. In other words, the jump by $L$ in is a combination of two jumps, by $p$ and $L-p$ (with $p$ random), each happening with a nonnegative probability.

In the second case $0<\nu\le\min\{\mu,\sqrt q\}$ the expression $\varphi_{q^{-1}, q^{g}, \mu \nu q^{g-1}}(p \mid \ell)$ might become negative, and the previous interpretation does not imply nonnegativity of $\mathbf{P}_{\ell,g}(L)$. Let us rewrite $\mathbf{P}_{\ell,g}(L)$ in two different ways (depending on the order of $\ell$ and $g$) to show the nonnegativity. We have (here and below in the proof we use the notation from ) $$\begin{aligned}
                \nonumber
        \mathbf{P}_{\ell,g}(L)
        &=
        \sum_{p=0}^{\min\{\ell,L\}}
        q^{gp}\frac{(\mu \nu/q;q^{-1})_{p}(q^{g};q^{-1})_{\ell-p}}{(\mu\nu q^{g-1};q^{-1})_{\ell}}
        \frac{(q^{-1};q^{-1})_{\ell}}{(q^{-1};q^{-1})_{p}(q^{-1};q^{-1})_{\ell-p}}
        \\&
        \nonumber
        \hspace{100pt}\times
        \mu^{L-p}
        \frac{(\nu\mu^{-1}q^p,\nu q^{g};q)_{L-p}}{(q,\nu^2 q^{g+p};q)_{L-p}}
    \frac{(\nu^2 q^{g+p},\mu;q)_{\infty}}{(\nu q^p,\mu\nu q^g;q)_{\infty}}
        \\&
        \label{P_ell_g_initial_formula}
        =
        \frac{(\mu; q)_{\infty}(\nu^{2} q^{g}; q)_{\infty}}{(\nu; q)_{\infty}(\nu \mu q^{g}; q)_{\infty}}
        \sum_{p=0}^{\min\{\ell,L\}}
        q^{gp}\frac{(\mu \nu/q;q^{-1})_{p}(q^{g};q^{-1})_{\ell-p}}{(\mu\nu q^{g-1};q^{-1})_{\ell}}
        \frac{(q^{-1};q^{-1})_{\ell}}{(q^{-1};q^{-1})_{p}(q^{-1};q^{-1})_{\ell-p}}
        \\&
        \hspace{100pt}\times
        \mu^{L-p}
        \frac{(\nu\mu^{-1}q^p,\nu q^{g};q)_{L-p}}{(q,\nu^2 q^{g+p};q)_{L-p}}
        \frac{(\nu;q)_p}{(\nu^2 q^g;q)_p}.
        \nonumber
    \end{aligned}$$ When $\ell<g$, this expression is rewritten as follows: $$\begin{gathered}
                \label{P_ell_g_rewrite_first_case}
        \mathbf{P}_{\ell,g}(L)
        =
        \mu^{L}\frac{(q^{g}; q^{-1})_{\ell} (\nu \mu^{-1};
        q)_{L}(\nu q^{g}; q)_{L}}{(\mu \nu q^{g-1}; q^{-1})_{\ell} (q;
        