Asymptotics of laurent polynomials of odd degree orthogonal with respect to varying exponential weights

Let Lambda R denote the linear space over R spanned by z(k), k is an element of Z. Define the ( real) inner product <center dot , center dot > L : Lambda(R) x Lambda(R) -> R, ( f, g) -> integral(R) f (s)g(s) exp(-NV(s)) ds, N is an element of N, where V satisfies: (i) V is real analytic on R\{0}; (ii) lim(|x|->infinity)(V(x)/ln(x(2) + 1)) = +infinity; and (iii) lim(|x|-> 0)(V(x)/ln(x(-2) + 1)) = +infinity. Orthogonalisation of the (ordered) base {1, z(-1), z, z(-2), z(2), . . . , z(,)(-k) z(k), . . .} with respect to <center dot , center dot > L yields the even degree and odd degree orthonormal Laurent polynomials {phi(m)(z)}(infinity) (m= 0) : phi 2n (z) = xi((2n))(-n) Z(-n) + . . .+ xi((2n))(n) Z(n) ,xi(n) ((2n)) > 0, and phi 2n+ 1(z) = xi((2n+ 1))(-n-1) Z(n-1) + . . . + xi((2n+1))(n) Z(n,) xi((2n-1))(-n) > 0 . Define the even degree and odd degree monic orthogonal Laurent polynomials: pi(2n)(z) := (xi((2n)-1)(n)phi 2n(Z) and pi 2n+1(Z) := (xi((2n+1)(-n-1))(-1) phi 2n+1(Z). Asymptotics in the double- scaling limit N, n ->infinity such that N/ n = 1 + o(1) of pi(2n+1)(Z) (in the entire complex plane),xi((2n+))(-n-1,) and xi(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].