A Uniform Asymptotic Formula for Orthogonal Polynomials Associated with exp(−x4)

Rui, Bo; Wong, R.

doi:10.1006/jath.1998.3282

Cited by 12 publications

(5 citation statements)

References 6 publications

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“…In our computations, we will only deal with the zero behavior in the positive real semi-axis, because the behavior in R − follows by reflection through the y-axis by symmetry reasons as usual. Thus, from (27), the positivity of K 2m−1 (0, 0; t), and Theorem 3 we are in the hypothesis of the Interlacing Lemma, and we immediately conclude the following results about monotonicity, asymptotics, and speed of convergence for the zeros of Q t 2m (x) in terms of the mass M . Let us define the monic polynomials…”

Section: Behavior and Monotonicity With M Of The Zeros Of Q Tsupporting

confidence: 69%

“…From now on, we will refer to this technique as the Interlacing Lemma. Here the linear combination of two polynomials of the same degree 2m is given by (27), and F t 2m , G 2m play the role of h n (x), g n (x) respectively. In order to apply this technique, we need to show that the hypotheses of the Interlacing Lemma are fulfilled.…”

Section: Behavior and Monotonicity With M Of The Zeros Of Q Tmentioning

confidence: 99%

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On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

Arceo

Huertas

Marcellán

2016

Applied Mathematics and Computation

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In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potentialWe analyze some properties of these polynomials, such as the ladder operators and the holonomic equation that they satisfy and, as an application, we give an electrostatic interpretation of their zero distribution in terms of a logarithmic potential interaction under the action of an external field. It is also shown that the coefficients of their three term recurrence relation satisfy a nonlinear difference string equation. Finally, an equation of motion for their zeros in terms of their dependence on t is given.

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Section: Behavior and Monotonicity With M Of The Zeros Of Q Tsupporting

confidence: 69%

Section: Behavior and Monotonicity With M Of The Zeros Of Q Tmentioning

confidence: 99%

On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

Arceo

Huertas

Marcellán

2016

Applied Mathematics and Computation

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“…which is (24). On the other hand, shifting the index n → 2n, and taking into account (22) we obtain (25).…”

Section: Propositionmentioning

confidence: 99%

On Freud–Sobolev type orthogonal polynomials

2019

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In this contribution we deal with sequences of monic polynomials orthogonal with respect to the Freud Sobolev-type inner productwhere p, q are polynomials, M 0 and M 1 are nonnegative real numbers. Connection formulas between these polynomials and Freud polynomials are deduced and, as a consequence, a five term recurrence relation for such polynomials is obtained. The location of their zeros as well as their asymptotic behavior is studied. Finally, an electrostatic interpretation of them in terms of a logarithmic interaction in the presence of an external field is given.

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“…Let P denote the linear space of real polynomials and consider the inner product p, q = R p(x)q(x)dµ(x), p, q ∈ P, (1) where µ is a symmetric positive measure supported in some (symmetric with respect to the origin) subset of R. It is easy to see that the associated sequence of monic orthogonal polynomials (MOPS in short) have a symmetry property, i.e., the even degree polynomials are even functions and the odd degree polynomials are odd functions. When dµ(x) = ω(x)dx = e −V(x) dx, where V(x) is an even polynomial with positive leading coefficient, the corresponding sequences of polynomials are the so-called Freud type orthogonal polynomials, currently the subject of intense analysis for several choices of the function V(x) (see, for example [1][2][3][4][5][6][7][8], among many others).…”

Section: Introductionmentioning

confidence: 99%

An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

Garza

Huertas

2021

Symmetry

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In this contribution we obtain some algebraic properties associated with the sequence of polynomials orthogonal with respect to the Sobolev-type inner product:p,qs=∫Rp(x)q(x)dμ(x)+M0p(0)q(0)+M1p′(0)q′(0), where p,q are polynomials, M0, M1 are non-negative real numbers and μ is a symmetric positive measure. These include a five-term recurrence relation, a three-term recurrence relation with rational coefficients, and an explicit expression for its norms. Moreover, we use these results to deduce asymptotic properties for the recurrence coefficients and a nonlinear difference equation that they satisfy, in the particular case when dμ(x)=e−x4dx.

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A Uniform Asymptotic Formula for Orthogonal Polynomials Associated with exp(−x4)

Cited by 12 publications

References 6 publications

On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

On polynomials associated with an Uvarov modification of a quartic potential Freud-like weight

On Freud–Sobolev type orthogonal polynomials

An Analysis of the Recurrence Coefficients for Symmetric Sobolev-Type Orthogonal Polynomials

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