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Asymptotics and Zeros of a Special F...

Lopez Santander, John J.,

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Asymptotics and Zeros of a Special Family of Jacobi Polynomials /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Asymptotics and Zeros of a Special Family of Jacobi Polynomials // John J Lopez Santander.
作者:	Lopez Santander, John J.,
面頁冊數:	1 electronic resource (91 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
標題:	Theoretical mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31939668
ISBN:	9798280700871

Asymptotics and Zeros of a Special Family of Jacobi Polynomials /
Lopez Santander, John J.,

Asymptotics and Zeros of a Special Family of Jacobi Polynomials /John J Lopez Santander. - 1 electronic resource (91 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

Classical Jacobi polynomials P(α,β)n (x), with α, β > −1, are well known for their orthogonality relation on the interval [−1, 1] with respect to the weight function w(x; α, β) = (1 − x)α (1 + x)β. In particular, this implies that all their zeros are distinct and lie in the open interval (−1, 1). By analytic continuation in the parameters α and β, Jacobi polynomials can be considered for general parameters α, β ∈ C. However, if Re(α) ≤ −1 or Re(β) ≤ −1, the orthogonality on [−1, 1] no longer holds, and the zeros are not necessarily real.This thesis uses the Riemann-Hilbert approach for the asymptotic analysis of a family of non-classical Jacobi polynomials with varying parameters of the form αn = n + 1/2 and βn = −n − 1/2 . For n ≥ 1, these polynomials lie outside the classical case. The main result is the derivation of global asymptotics for these polynomials, which is then used to establish the location of their zeros. The analysis is based on the Riemann-Hilbert formulation of Jacobi polynomials derived from the non-Hermitian orthogonality introduced by Kuijlaars et al. This family of polynomials arises in the symbolic evaluation of integrals in the work of Boros and Moll, and corresponds to a limiting case not considered in the works of Kuijlaars et al. A remarkable feature of the analysis emerges in the local study of the Riemann- Hilbert problem near the origin, where the local parametrix introduces a pole. This requires a special analysis to obtain an analytic solution, which is then expressed in terms of the parabolic cylinder function.

English

ISBN: 9798280700871Subjects--Topical Terms:

3173530
Theoretical mathematics.
Subjects--Index Terms:

Asymptotics

Asymptotics and Zeros of a Special Family of Jacobi Polynomials /
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502 $b Ph.D. $c Tulane University $d 2025.
520 $a Classical Jacobi polynomials P(α,β)n (x), with α, β > −1, are well known for their orthogonality relation on the interval [−1, 1] with respect to the weight function w(x; α, β) = (1 − x)α (1 + x)β. In particular, this implies that all their zeros are distinct and lie in the open interval (−1, 1). By analytic continuation in the parameters α and β, Jacobi polynomials can be considered for general parameters α, β ∈ C. However, if Re(α) ≤ −1 or Re(β) ≤ −1, the orthogonality on [−1, 1] no longer holds, and the zeros are not necessarily real.This thesis uses the Riemann-Hilbert approach for the asymptotic analysis of a family of non-classical Jacobi polynomials with varying parameters of the form αn = n + 1/2 and βn = −n − 1/2 . For n ≥ 1, these polynomials lie outside the classical case. The main result is the derivation of global asymptotics for these polynomials, which is then used to establish the location of their zeros. The analysis is based on the Riemann-Hilbert formulation of Jacobi polynomials derived from the non-Hermitian orthogonality introduced by Kuijlaars et al. This family of polynomials arises in the symbolic evaluation of integrals in the work of Boros and Moll, and corresponds to a limiting case not considered in the works of Kuijlaars et al. A remarkable feature of the analysis emerges in the local study of the Riemann- Hilbert problem near the origin, where the local parametrix introduces a pole. This requires a special analysis to obtain an analytic solution, which is then expressed in terms of the parabolic cylinder function.
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590 $a School code: 0235
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical physics. $3 2144760
653 $a Asymptotics
653 $a Non-classical jacobi polynomials
653 $a Riemann-Hilbert problems
653 $a Zeros
653 $a Orthogonality relation
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690 $a 0753
690 $a 0405
690 $a 0364
710 2 $a Tulane University. $b Mathematics. $e degree granting institution. $3 3819232
720 1 $a Moll, Victor $e degree supervisor.
720 1 $a McLaughlin, Kenneth D. T-R $e degree supervisor.
773 0 $t Dissertations Abstracts International $g 86-12B.
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791 $a Ph.D.
792 $a 2025
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