$$\label{qmunn}
q_{\mu_n,n}= \left(1+\mathcal{O}(e^{-c n})\right) \exp\left\{ -n V^{\overline{\nu}}\right\}K_{1,n}+\mathcal{O}(e^{-c n})\exp\left\{- n V^{\overline{\nu}}\right\} K_{2,n}$$ and $$\label{qmunn-1}
 \frac{d_{n,n-1}}{2^{2 n a}} q_{\mu_n,n-1}=\left(1+\mathcal{O}(e^{-c n})\right) \exp\left\{ -n V^{\overline{\nu}}\right\}K_{2,n}+\mathcal{O}(e^{-c n})\exp\left\{- n V^{\overline{\nu}}\right\} K_{1,n}$$ where $\displaystyle d_{n,n-1}=-\left(2 \pi i \left|\left|q_{\mu_n,n-1}\right|\right|^2_{\mu_n,2}\right)^{-1},$ $$\label{forK1}
K_{1,n}(x)=\displaystyle 2  \cos n\left((1-a)\pi \int^1_x \mbox{d} \widetilde{\sigma}(t) -a \arccos{x} \right),$$ and $$\label{forK2}
K_{2,n}(x)=\displaystyle  \frac{1}{i} \cos n \left((1-a) \pi \int_{x}^1 \mbox{d} \widetilde{\sigma}(t) - (a-1/n) \arccos{x} \right).$$

We study a matrix Riemann-Hilbert problem like in [@KMVV Theorem 2.4] whose solution $Y$ is a $2 \times 2$ matrix function satisfying the following conditions:

1.  $Y \in \mathcal{H}(\mathbb{C}\setminus[-1,\,1])$ (all the entries of $Y$ are analytic on $\mathbb{C}\setminus[-1,\,1]$),

2.  $\displaystyle Y_+(x)=Y_-(x)\left(\begin{array}{c c}
    1 & \left(q_{m(n)}(x)\sqrt{1-x^2}\right)^{-1}\\ & \\
    0 & 1
    \end{array}\right)$, $x \in (-1,\,1)$,

3.  $\displaystyle Y(z)\left(\begin{array}{c c}
    z^{-n} & 0\\ & \\
    0 & z^{n}
    \end{array}\right)=\mathbb{I}+\mathcal{O}(1/z)$ as $z \to \infty$, $\mathbb{I}$ is the $2\times2$ identity matrix.

4.  $\displaystyle Y(z)=\mathcal{O}\left(\begin{array}{c c}
    1 & |z\pm 1|^{-1/2} \\ & \\
    1 &  |z\pm 1|^{-1/2}
    \end{array}\right)$ as $z \to \mp 1$.

According to [@KMVV Theorem 2.4] (see also [@Ku]) the $Y$ solution of above matrix Riemann-Hilbert problem (for short Y-RHP) is unique and has the form $$Y(z)=\left(\begin{array}{c c}
q_{\mu_n,n}(z) & \displaystyle -\frac{1}{2 \pi i}\int \frac{q_{\mu_n,n}(x)}{z-x} \, \mbox{d}\mu_n(x) \\ & \\
d_{n,n-1} q_{\mu_n,n-1}(z) & \displaystyle -\frac{d_{n,n-1}}{2 \pi 