Asymptotics of laurent polynomials of odd degree 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 Ċ,Ċ L : Λ ^ℝ× Λ ^ℝ ℝ, (f,g) ∫_ℝ f(s)g(s) exp(- N V(s)) ds, N ℕ, where V satisfies: (i) V is real analytic on ℝ 0; (ii) lim x (V(x)/ln(x²} + 1)) = + and (iii) limx 0(V(x)/ln (x²} + 1)) = +. Orthogonalisation of the (ordered) base 1,z^-1,z,z^-2z²},z ^-k},z^k with respect to , {{ L}} yields the even degree and odd degree orthonormal Laurent polynomials φ{m}(z)_m=0: φ_2n(z) = ξ⁽²ⁿ⁾z^-n + + ξ⁽²ⁿ⁾_nzⁿ ξ⁽²ⁿ⁾_n > 0, and φ^{2n+1}(z) = ξ⁽²ⁿ⁺¹⁾ _-n-1z^-n-1 + + ξ^(2n+1)nz ⁿ ξ⁽²ⁿ⁺¹⁾_-n-1 > 0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π_2n(z) := (ξ⁽²ⁿ⁾_n^-1} φ_2n(z) and π^{2n+1}(z) := (ξ⁽²ⁿ⁺¹⁾_-n-1^-1 φ_2n+1(z). Asymptotics in the double-scaling limit N,n such that N,n = 1 + o(1) of π_2n+1(z) (in the entire complex plane), ξ⁽²ⁿ⁺¹⁾_-n-1, and φ_2n+1(z)(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on ℝ, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].