Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights

Let Λ^ℝ denote the linear space over ℝ spanned by z ^k , k ∈ ℤ. Define the real inner product 〈 .,.〉 _L : Λ^ℝ×Λ^ℝ→ℝ, (f,g)∫_ℝ}f(s)g(s)exp (-{N}V(s)){d}s, N ∈, where V satisfies: (i) V is real analytic on ℝ/{0}; (ii) lim∈ _{| x |→∞}(V(x)/ln∈(x ²+1))=+∞; and (iii) lim∈ _{| x |→0}(V(x)/ln∈(x ^-2+1))= +∞. Orthogonalisation of the (ordered) base with respect to 〈 .,.〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ _2n (z)= _k=-n ⁿ ξ _k ⁽²ⁿ⁾ z ^k , ξ _n ⁽²ⁿ⁾ >0, and φ _2n+1(z)= _k=-n-1 ⁿ ξ _k ⁽²ⁿ⁺¹⁾ z ^k , ξ _-n-1 ⁽²ⁿ⁺¹⁾ >0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c _2n ^# φ _2n-2(z)+b _2n ^# φ _2n-1(z)+a _2n ^# φ _2n (z)+b _2n+1 ^# φ _2n+1(z)+c _2n+2 ^# φ _2n+2(z) and z φ _2n+1(z)=b _2n+1 ^# φ _2n (z)+a _2n+1 ^# φ _2n+1(z)+b _2n+2 ^# φ _2n+2(z), where c ₀ ^# =b ₀ ^# =0, and c _2k ^# >0, k ∈, and z ^-1 φ _2n+1(z)=γ _2n+1 ^# φ _2n-1(z)+β _2n+1 ^# φ _2n (z)+α _2n+1 ^# φ _2n+1(z)+β _2n+2 ^# φ _2n+2(z)+γ _2n+3 ^# φ _2n+3(z) and z ^-1 φ _2n (z)=β _2n ^# φ _2n-1(z)+α _2n ^# φ _2n (z)+β _2n+1 ^# φ _2n+1(z), where β ₀ ^# =γ ₁ ^# =0, β ₁ ^# >0, and γ _2l+1 ^# >0, l ∈. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence k= ∫_ℝ, 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 ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295-368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277-337, [1995]) and (Int. Math. Res. Not. 6:285-299, [1997]).