Monotonicity of zeros of orthogonal Laurent polynomials

Dimitrov, Dimitar K.; Ranga, A. Sri

doi:10.4310/maa.2002.v9.n1.a1

Cited by 7 publications

(5 citation statements)

References 14 publications

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“…which is the L-analogue of Jacobi polynomials or orthogonal Laurent Jacobi polynomials considered in [15]. It is important to remark some recent contributions especially [25,26] regarding distribution of zeros of q-Leguerre polynomials and some new hypergeometric type functions.…”

Section: Behaviour Of Zerosmentioning

confidence: 99%

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Rational spectral transformation of continued fractions associated to a perturbed $R_I$ type recurrence relations

Shukla¹,

Swaminathan²

2022

Preprint

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In this work, orthogonal polynomials satisfying recurrence relation, with P −1 (z) = 0 and P 0 (z) = 1 are analyzed when modifications of the recurrence coefficient is considered. Specifically, representation of new perturbed polynomials in terms of old unperturbed ones, behaviour of zeros and spectral transformation of Stieltjes function are given. Further, Toda lattice equations corresponding to perturbed system of recurrence coefficients are obtained. Finally, when λ n is a positive chain sequence, co-dilation and its consequences are interpreted with the help of some illustrations.

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Section: Behaviour Of Zerosmentioning

confidence: 99%

“…(x) = 1, we get the monic polynomials It is shown [15] that c n and λ n are positive when c > a > N or a < c < 1 − N, n = 1, 2, . .…”

Section: Behaviour Of Zerosmentioning

confidence: 99%

Rational spectral transformation of continued fractions associated to a perturbed $R_I$ type recurrence relations

Shukla¹,

Swaminathan²

2022

Preprint

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“…The paradigm for the asymptotic (as n → ∞) analysis of the respective (matrix) RHPs is a union of the Deift-Zhou (DZ) non-linear steepest-descent method [1,2], used for the asymptotic analysis of undulatory RHPs, and the extension of Deift-Venakides-Zhou [3], incorporating into the DZ method a non-linear analogue of the WKB method, making the asymptotic analysis of fully non-linear problems tractable (it should be mentioned that, in this context, the equilibrium measure [43] plays an absolutely crucial rôle in the analysis [44]); see, also, the multitudinous extensions and applications of the DZ method . It is worth mentioning that asymptotics for Laurent-type polynomials and their zeros have been obtained in [33,70] (see, also, [71][72][73]).…”

Section: Introductionmentioning

confidence: 99%

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

2007

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Let Λ R denote the linear space over R spanned by z k , k ∈ Z. Define the real inner product (with varying exponential weights)Define the even degree and odd degree monic orthogonal Laurent polynomials: π π π 2n (z) := (ξ (2n) n ) −1 φ 2n (z) and π π π 2n+1 (z) := (ξ (2n+1) −n−1 ) −1 φ 2n+1 (z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1+o(1) of π π π 2n+1 (z) (in the entire complex plane), ξ (2n+1) −n−1 , φ 2n+1 (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequencek∈Z are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the largen behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2,3].

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“…15] and proved in [5, Thm. 7.1.1] (see also [23,Thm. 1]) can be applied to discrete orthogonal polynomials such as Meixner and Hahn polynomials as well as orthogonal Laurent polynomials.…”

Section: Introductionmentioning

confidence: 99%

Monotonicity of zeros of polynomials orthogonal with respect to an even weight function

Jordaan

Wang

Zhou

2014

Integral Transforms and Special Functions

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The monotonicity properties of all the zeros with respect to a parameter of orthogonal polynomials associated with an even weight function are studied. The results we obtain extend the work of A. Markoff. The monotonicity of the zeros of Gegenbauer, Freud-type and symmetric Meixner-Pollaczek orthogonal polynomials as well as Al-Salam-Chihara q-orthogonal polynomials are investigated. For the Meixner-Pollaczek polynomials, a special case of a conjecture by Jordaan and Toókos which concerns the interlacing of their zeros between two different sequences of Meixner-Pollaczek polynomials is proved.

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Monotonicity of zeros of orthogonal Laurent polynomials

Cited by 7 publications

References 14 publications

Rational spectral transformation of continued fractions associated to a perturbed $R_I$ type recurrence relations

Rational spectral transformation of continued fractions associated to a perturbed $R_I$ type recurrence relations

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

Monotonicity of zeros of polynomials orthogonal with respect to an even weight function

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