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Let   denote the linear space over   spanned by   . Define the (real) inner product   , where V satisfies: (i) V is real analytic on   ; (ii)  ; and (iii)  . Orthogonalisation of the (ordered) base   with respect to   yields the even degree and odd degree orthonormal Laurent polynomials   0$" align="middle" border="0"> , and   0$" align="middle" border="0"> . Define the even degree and odd degree monic orthogonal Laurent polynomials:   and   . Asymptotics in the double-scaling limit   such that   of   (in the entire complex plane),   , and   (in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on   , and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2],[3].

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights