Abstract
Let \(\Lambda^{{\Bbb R}}\) denote the linear space over \({\Bbb R}\) spanned by \(z^{k}, k \in {\Bbb Z}\). Define the (real) inner product \(\langle \cdot,\cdot \rangle_{{\cal L}} : \Lambda^{{\Bbb R}} \times \Lambda^{{\Bbb R}} \to {\Bbb R}, (f,g) \mapsto \int_{{\Bbb R}}f(s)g(s) \exp (-{\cal N} V(s)) \, {\rm d} s, {\cal N} \in {\Bbb N}\), where V satisfies: (i) V is real analytic on \({\Bbb R} \backslash \{0\}\); (ii)\(\lim_{\vert x \vert \to \infty}(V(x)/{\rm ln} (x^{2} + 1)) = +\infty\); and (iii)\(\lim_{\vert x \vert \to 0}(V(x)/{\rm ln} (x^{-2} + 1)) = +\infty\). Orthogonalisation of the (ordered) base \(\lbrace 1,z^{-1},z,z^{-2},z^{2},\ldots,z^{-k},z^{k},\ldots \rbrace\) with respect to \(\langle \cdot,\cdot \rangle_{{\cal L}}\) yields the even degree and odd degree orthonormal Laurent polynomials \(\lbrace \phi_{m}(z) \rbrace_{m=0}^{\infty}: \phi_{2n}(z) = \xi^{(2n)}_{-n}z^{-n} + \cdots + \xi^{(2n)}_{n}z^{n}, \xi^{(2n)}_{n} > 0\), and \(\phi_{2n+1}(z) = \xi^{(2n+1)}_{-n-1}z^{-n-1} + \cdots + \xi^{(2n+1)}_{n}z^{n}, \xi^{(2n+1)}_{-n-1} > 0\). Define the even degree and odd degree monic orthogonal Laurent polynomials: \(\pi_{2n}(z) := (\xi^{(2n)}_{n})^{-1} \phi_{2n}(z)\) and \(\pi_{2n+1}(z) := (\xi^{(2n+1)}_{-n-1})^{-1} \phi_{2n+1}(z)\). Asymptotics in the double-scaling limit \({\cal N},n \to \infty\) such that \({\cal N}/n = 1 + o(1)\) of \(\pi_{2n+1}(z)\) (in the entire complex plane), \(\xi^{(2n+1)}_{-n-1}\), and \(\phi_{2n+1}(z)\) (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on \({\Bbb R}\), and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].
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McLaughlin, K., Vartanian, A. & Zhou, X. Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights. Constr Approx 27, 149–202 (2008). https://doi.org/10.1007/s00365-007-0675-z
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DOI: https://doi.org/10.1007/s00365-007-0675-z