Math @ Duke

Publications [#244277] of Xin Zhou
Papers Published
 McLaughlin, KTR; Vartanian, AH; Zhou, X, Asymptotics of recurrence relation coefficients, hankel determinant ratios, and root products associated with laurent polynomials orthogonal with respect to varying exponential weights,
Acta Applicandae Mathematicae, vol. 100 no. 1
(2008),
pp. 39104, ISSN 01678019 [doi]
(last updated on 2018/04/23)
Abstract: 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=n1n ξ k(2n+1) z k , ξ n1(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 2n2(z)+b 2n# φ 2n1(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 2n1(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 2n1(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 doublescaling 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 realvalued, biinfinite 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 RiemannHilbert problems on ℝ, and then extracting the largen behaviours by applying the nonlinear steepestdescent method introduced in (Ann. Math. 137(2):295368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277337, [1995]) and (Int. Math. Res. Not. 6:285299, [1997]). © 2007 Springer Science+Business Media B.V.


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