: $$\label{push_duality_formulation_equation_detail}
    \begin{split}
        &
        \mu^k
        \sum_{\vec x'\in \mathbb{X}^{N}}
        \sum_{\vec y'\in \mathbb{Y}^{N}_k}
        \,\prod_{i=1}^N
        \mathbf{P}_{x_{i-1}-x_{i-1}',x_{i-1}-x_i-1}(x_i-x_i')
        \prod_{i=1}^{N}
        \varphi_{q,q/\mu,\nu}
        \left( \sum\nolimits_{j=0}^{i-1}(y_j'-y_j)\;\Big\vert\;y_i \right)
        \prod_{r=0}^{N}q^{y_r'(x_r'+r)}
        \\
        &\hspace{120pt}
        =
        \nu^k
        \sum_{\vec y''\in \mathbb{Y}^{N}_k}
        \,
        \prod_{i=1}^{N}\varphi_{q,q/\nu,\nu}
        \left( \sum\nolimits_{j=0}^{i-1}(y_j''-y_j)\;\Big\vert\;y_i \right)
        \prod_{r=0}^{N}q^{y_r''(x_r+r)}
    \end{split}$$ (the products over $r$ in both sides vanish if $y_0'$ or $y_0''$, respectively, are positive). We will prove by induction on $N$.

\[lemma:induciton\_base\] For any $x_1\in \mathbb{Z}$ and $y_1\ge0$ there exists $\mu_0>0$ such that for all $0<\mu<\mu_0$ we have $$\mu^{y_1}\varphi_{q,q/\mu,\nu}(0\mid y_1)
        \sum_{x_1'=-\infty}^{x_1}\varphi_{q,\mu,\nu}(x_1-x_1'\mid \infty)
        \,q^{y_1(x_1'+1)}
        =
        \nu^{y_1}\varphi_{q,q/\nu,\nu}(0\mid y_1)\,
        q^{y_1(x_1+1)}.$$

Expand the definition of $\varphi$ , , and rewrite the claim as $$\mu^{y_1}
        \frac{(q/\mu;q)_{y_1}}{(\nu;q)_{y_1}}
        \sum_{x_1'=-\infty}^{x_1}
        \mu^{x_1-x_1'}
        \frac{(\nu/\mu;q)_{x_1-x_1'}}{(q;q)_{x_1-x_1'}}
        \frac{(\mu;q)_{\infty}}{(\nu;q)_{\infty}}
        \,q^{y_1(x_1'+1)}
        =
        \nu^{y_1}
        \frac{(q/\nu;q)_{y_1}}{(\nu;q)_{y_1}}\,
        q^{y_1(x_1+1)}.$$ This simplifies to $$\sum_{d=0}^{\infty}
        (\mu q^{-y_1})^{d}
        \frac{(\nu/\mu;q)_{d}}{(q;q)_{d}}
        \frac{(\mu q^{-y_1};q)_{\infty}}{(\nu q^{-y_1};q)_{\infty}}
        =1,\qquad d=x_1-x_1',$$ which is simply $\sum_{d=0}^{\infty}\varphi_{q,\mu q^{-y_1},\nu q^{-y_1}}(d\mid\infty)=1$. Note that the series converges for sufficiently small $\mu$.

The induction step $N-1\to N$ is based on the following lemma:

\[lemma:main\_identity\] For all nonnegative integers $g,\ell,y$ there exists $\mu_0>0$ such that for all $0<\mu<\mu_0$ we have $$\label{main_identity_equation}
        \begin{split}
            &
            \sum_{t=0}^{y}
            \frac{(q; q)_{y}}{(q; q)_{t}(q; q)_{y-t}}\,
            q^{(g+1)t}
            \nu^{y-t} (\nu^{-1}q; q)_{y-t}
            (\nu^{2}q^{-1}; q)_{t}
            \\&
            \hspace{50pt}
            =
            \sum_{s=0}^{y}
            \frac{(q; q)_{y}}{(q; q)_{s}(q; q)_{y-s}}\,
            q^{(g+1-\ell)s}
            \mu^{y-s}
            (\mu^{-1}q; q)_{y-s}
            (\mu \nu q^{-1}; q)_{s}
            \sum_{L=0}^{\infty} q^{-L(y-s)} \mathbf{P}_{\ell, g}(L).
        \end{split}$$

We prove in the next .

Denote 