Asymptotics of Recurrence Relation Coefficients, Hankel Determinant Ratios, and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Abstract

Let R denote the linear space over R spanned by z k , k ∈ Z. Define the real inner product ·, · L :. . } with respect to ·, · L yields the even degree and odd degree orthonormal Laurent polynomialsAssociated with the even degree and odd degree OLPs are the following two pairs of recurrence rela-, where β 0 = γ 1 = 0, β 1 > 0, and γ 2l+1 > 0, l ∈ N. 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 {c k = R s k exp(−N V (s)) d s} k∈Z , 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 large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295-368, 1993) and further de-

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

“…In order to obtain asymptotics (as n → ∞) for ξ (2n+1) Even though the results of Lemma A.1 below, namely, large-z asymptotics (as (C \ R ∋) z → ∞) of o Y(z), are not necessary in order to prove Theorems 2.3.1 and 2.3.2, they are essential for the results of [40], related to asymptotics of the coefficients of the system of three-and five-term recurrence relations and the corresponding Laurent-Jacobi matrices (cf. Section 1).…”

“…(iii) (iii) (iii) in Part III [40], asymptotics (as n → ∞) of ν ) −1 ), as well as of the (elements of the) Laurent-Jacobi matrices, J and K , and other, related, quantities constructed from the coefficients of the three-and five-term recurrence relations, are obtained.…”

“…, n, and φ 2n+1 (z) = ξ (2n+1) −n−1 (z)π π π 2n+1 (z), one proceeds as follows: 2πi(ξ (2n+1) −n−1 ) 2 R φ 2n+1 (s)φ 2n+1 (s)e −n V(s) ds = above two asymptotic expansions for ( o Y(z)z nσ 3 ) 12 , one arrives at the result stated in the Proposition. Using the results of Propositions 5.3 and 5.4, one obtains n → ∞ asymptotics for ξ (2n+1) −n−1 and φ 2n+1 (z) (in the entire complex plane) stated in Theorem 2.3.2.Large-z asymptotics for o Y(z) are given in the Appendix (see Lemma A.1): these latter asymptotics are necessary for the results of[40].…”

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

“…1 , thus φ 2n+1 (z) (cf. Equation (1.5)), and the Hankel determinant ratio H in Part III[52], asymptotics (as n → ∞) of ν…”

“…Even though the results of Lemma A.1 below, namely, small-z asymptotics (as (C \ R ∋) z → 0) of e Y(z), are not necessary in order to prove Theorems 2.3.1 and 2.3.2, they are essential for the results of [52], related to asymptotics of the coefficients of the system of three-and five-term recurrence relations and the corresponding Laurent-Jacobi matrices (cf. Section 1).…”