+ b(z) \psi_b(z)$$ and we have, since ${\lvert\Gamma_+\rvert} > {\lvert\Gamma_-\rvert}$ on ${{\mathbb{C}}}\backslash\text{bands}$:

[T3.2]{} For $z\in{{\mathbb{C}}}\backslash\text{bands}$, $${\label}{3.33} 
\lim_{m\to\infty}\, z^{-mp/2} \Gamma_+^{-m} \varphi_{mp+b}(z) = j_b(z)$$ In addition, $j_b$ is nonvanishing near $z=\infty$.

In particular, if $z_0\notin\text{bands}$ and $j_b (z_0)\neq 0$, then for some ${\varepsilon}>0$ and $M$, we have $\varphi_{mp+b}(z_0)\neq 0$ if ${\lvertz-z_0\rvert}<{\varepsilon}$ and $m\geq M$. If $z_0\notin\text{bands}$ and $j_b(z_0)$ has a zero of order $k$, then for some ${\varepsilon}>0$ and all $m$ large, $\varphi_{mp+b}(z)$ has precisely $k$ zeros [[(]{}]{}counting multiplicity[[)]{}]{}. The number of $z_0$ in ${{\mathbb{C}}}\backslash\text{bands}$ with $j_b(z_0)=0$ is at most $2p + 2b-1$.

As noted, and ${\lvert\Gamma_+\rvert} > {\lvert\Gamma_-\rvert}$ imply . To analyze $j_b(z)$ near $z=\infty$, we proceed as follows: We have, by and , that as ${\lvertz\rvert}\to\infty$, $$\begin{aligned}
T_p(z) &= z^p \biggl( \, \prod_{j=0}^{p-1} \rho_j^{-1}\biggr) 
\biggl[ \begin{pmatrix} 1 & 0 \\ -\alpha_{p-1} & 0 \end{pmatrix} \dots 
\begin{pmatrix} 1 & 0 \\ -\alpha_0 & 0 \end{pmatrix} \biggr] + O(z^{p-1}) {\label}{3.34} \\
&= z^p \biggl(\, \prod_{j=1}^{p-1} \rho_j^{-1} \biggr) 
\begin{pmatrix} 1 & 0 \\ -\alpha_{p-1} & 0 \end{pmatrix} + O(z^{p-1}) {\label}{3.35} \end{aligned}$$ from which it follows that $$\begin{aligned}
P_+ &= \begin{pmatrix} 1 & 0 \\ -\alpha_{p-1} & 0 \end{pmatrix} + O(z^{-1}) {\label}{3.36} \\ 
P_- &= \begin{pmatrix} 0 & 0 \\ \alpha_{p-1} & 1 \end{pmatrix} 