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Publications [#244277] of Xin Zhou

Papers Published

  1. 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 (January, 2008), pp. 39-104, Springer Nature, ISSN 0167-8019 [doi]
    (last updated on 2021/05/15)

    Abstract:
    Let Λ denote the linear space over ℝ spanned by z , k ∈ ℤ. Define the real inner product 〈 .,.〉 : Λ ×Λ →ℝ, (f,g)∫ }f(s)g(s)exp (-{N}V(s)){d}s, N ∈, where V satisfies: (i) V is real analytic on ℝ/{0}; (ii) lim∈ (V(x)/ln∈(x +1))=+∞; and (iii) lim∈ (V(x)/ln∈(x +1))= +∞. Orthogonalisation of the (ordered) base with respect to 〈 .,.〉 yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ (z)= ξ z , ξ >0, and φ (z)= ξ z , ξ >0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ (z)=c φ (z)+b φ (z)+a φ (z)+b φ (z)+c φ (z) and z φ (z)=b φ (z)+a φ (z)+b φ (z), where c =b =0, and c >0, k ∈, and z φ (z)=γ φ (z)+β φ (z)+α φ (z)+β φ (z)+γ φ (z) and z φ (z)=β φ (z)+α φ (z)+β φ (z), where β =γ =0, β >0, and γ >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= ∫ , 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. ℝ k ℝ ℝ 2 -2 n (2n) k (2n) n (2n+1) k (2n+1) # # # # # # # # # # # -1 # # # # # -1 # # # # # # # L ℝ | x |→∞ | x |→0 L 2n k=-n k n 2n+1 k=-n-1 k -n-1 2n 2n 2n-2 2n 2n-1 2n 2n 2n+1 2n+1 2n+2 2n+2 2n+1 2n+1 2n 2n+1 2n+1 2n+2 2n+2 0 0 2k 2n+1 2n+1 2n-1 2n+1 2n 2n+1 2n+1 2n+2 2n+2 2n+3 2n+3 2n 2n 2n-1 2n 2n 2n+1 2n+1 0 1 1 2l+1 ℝ

 

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