Recurrence coefficients of Toda-type orthogonal polynomials I. Asymptotic analysis

We define the family of truncated Laguerre polynomials Pn(x; z), orthogonal with respect to the linear functional ℓ defined by ⟨ℓ, p⟩ = z 0 p(x)x α e −x dx, α > −1. The connection between Pn(x; z) and the polynomials Sn(x; z) (obtained through the symmetrization process) constitutes a key element in our analysis. As a consequence, several properties of the polynomials Pn(x; z) and Sn(x; z) are studied taking into account the relation between the parameters of the three-term recurrence relations that they satisfy. Asymptotic expansions of these coefficients are given. Discrete Painlevé and Painlevé equations associated with such coefficients appear in a natural way. An electro-static interpretation of the zeros of such polynomials as well as the dynamics of the zeros in terms of the parameter z are given. MSC Classification: Primary: 42C05; 33C50

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“…as n → ∞. As it was observed in [17], the coefficients s n,k (after some k) form an asymptotic sequence as n → ∞.…”

Section: Asymptotic Analysissupporting

confidence: 55%

Symmetrization process and truncated orthogonal polynomials

Dominici

García-Ardila

Marcellán

2023

Preprint

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“…(∆ + ∇) η n,j η n,k−j , k ≥ 2, and it follows that (88) is an asymptotic series as n → ∞. Using the same methods that we introduced in [7], we can show that for k ≥ 2…”

Section: Theorem 27mentioning

confidence: 74%

“…The (generally nonlinear) equations satisfied by the coefficients of the threeterm recurrence relation (7) are known in the literature as Laguerre-Freud equations (see [4]). They can be consider discrete analogues of the Painlevé equations [19].…”

Section: Propositionmentioning

confidence: 99%