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

Let Λℝ denote the linear space over ℝ spanned by z k , k ∈ ℤ. Define the real inner product 〈 .,.〉 L : Λℝ×Λℝ→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=-nn ξ k(2n) z k , ξ n(2n) >0, and φ 2n+1(z)= k=-n-1n 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]). © 2007 Springer Science+Business Media B.V.

Duke Scholars

Author Xin Zhou Mathematics