  Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2},
>...c,z^{-k},z^{k},...c \rbrace$ with respect to the real inner product $(f,g)
\mapsto \int_{\mathbb{R}}f(s)g(s) \exp (-\mathscr{N} V(s)) \md s$, $\mathscr{N}
\in \mathbb{N}$, where $V$ is real analytic on $\mathbb{R} \setminus \{0\}$,
$\lim_{| x | \to \infty}(V(x)/ \ln (x^{2}
  + 1)) = +\infty$, and $\lim_{| x | \to 0} (V(x)/\ln (x^{-2}
  + 1)) = +\infty$, yields the even degree and odd degree orthonormal Laurent
polynomials (OLPs), $\phi_{2n}(z) = \sum_{k=-n}^{n} \xi^{(2n)}_{k}z^{k}$, with
$\xi^{(2n)}_{n} > 0$, and $\phi_{2n+1}(z) =
  \sum_{k=-n-1}^{n} \xi^{(2n+1)}_{k}z^{k}$, with $\xi^{(2n+1)}_{-n-1} > 0$,
respectively. Associated with the even degree and odd degree OLPs are two pairs
of three- and five-term recurrence relations. Asymptotics in the double-scaling
limit as $\mathscr{N},n \to \infty$ such that $\mathscr{N}/n = 1 + o(1)$ of the
coefficients of these two pairs of recurrence relations, Hankel determinant
ratios, and the products of the (real) roots of the OLPs are obtained by
formulating the even degree and odd degree OLP problems as matrix
Riemann-Hilbert problems on $\mathbb{R}$, and then extracting the large-N
behaviours by applying the non-linear steepest-descent method introduced in [1]
and further developed in [2,3].
