A note on the distance between two consecutive zeros of m-orthogonal polynomials for a generalized Jacobi weight

We consider the generalized Jacobi weight [Formula: see text], [Formula: see text]. As is shown in [D. Dai and L. Zhang, Painlevé VI and Henkel determinants for the generalized Jocobi weight, J. Phys. A: Math. Theor. 43 (2010), Article ID:055207, 14pp.], the corresponding Hankel determinant is the [Formula: see text]-function of a particular Painlevé VI. We present all the possible asymptotic expansions of the solution of the Painlevé VI equation near [Formula: see text] and [Formula: see text] for generic [Formula: see text]. For four special cases of [Formula: see text] which are related to the dimension of the Hankel determinant, we can find the exceptional solutions of the Painlevé VI equation according to the results of [A. Eremenko, A. Gabrielov and A. Hinkkanen, Exceptional solutions to the Painlevé VI equation, preprint (2016), arXiv:1602.04694 ], and thus give another characterization of the Hankel determinant.

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Mehler–Heine formulas for orthogonal polynomials with respect to the modified Jacobi weight

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-2014-11976-9 ◽

2014 ◽

Vol 142 (6) ◽

pp. 2035-2045 ◽

Cited By ~ 3

Author(s):

Bujar Xh. Fejzullahu

Keyword(s):

Orthogonal Polynomials ◽

Jacobi Weight

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Jacobi-weighted orthogonal polynomials on triangular domains

Journal of Applied Mathematics ◽

10.1155/jam.2005.205 ◽

2005 ◽

Vol 2005 (3) ◽

pp. 205-217 ◽

Cited By ~ 6

Author(s):

A. Rababah ◽

M. Alqudah

Keyword(s):

Weight Function ◽

Orthogonal Polynomials ◽

Bernstein Polynomial ◽

Orthogonal System ◽

Polynomial Basis ◽

Bernstein Basis ◽

Triangular Domain ◽

Jacobi Weight ◽

Bernstein Polynomial Basis ◽

Alpha Beta

We construct Jacobi-weighted orthogonal polynomials𝒫n,r(α,β,γ)(u,v,w),α,β,γ>−1,α+β+γ=0, on the triangular domainT. We show that these polynomials𝒫n,r(α,β,γ)(u,v,w)over the triangular domainTsatisfy the following properties:𝒫n,r(α,β,γ)(u,v,w)∈ℒn,n≥1,r=0,1,…,n,and𝒫n,r(α,β,γ)(u,v,w)⊥𝒫n,s(α,β,γ)(u,v,w)forr≠s. And hence,𝒫n,r(α,β,γ)(u,v,w),n=0,1,2,…,r=0,1,…,nform an orthogonal system over the triangular domainTwith respect to the Jacobi weight function. These Jacobi-weighted orthogonal polynomials on triangular domains are given in Bernstein basis form and thus preserve many properties of the Bernstein polynomial basis.

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Asymptotics of Sobolev orthogonal polynomials for a Jacobi weight

Methods and Applications of Analysis ◽

10.4310/maa.1997.v4.n4.a4 ◽

1997 ◽

Vol 4 (4) ◽

pp. 430-437

Author(s):

Andrei Martínez-Finkelshtein ◽

Juan J. Moreno-Balcázar

Keyword(s):

Orthogonal Polynomials ◽

Jacobi Weight ◽

Sobolev Orthogonal Polynomials

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ONSAGER'S ALGEBRA AND PARTIALLY ORTHOGONAL POLYNOMIALS

International Journal of Modern Physics B ◽

10.1142/s0217979202011883 ◽

2002 ◽

Vol 16 (14n15) ◽

pp. 2129-2136 ◽

Cited By ~ 10

Author(s):

G. VON GEHLEN

Keyword(s):

Weight Function ◽

Orthogonal Polynomials ◽

Potts Model ◽

Jacobi Polynomials ◽

Chiral Potts Model ◽

Recursion Relations ◽

Energy Eigenvalues ◽

Special Polynomials ◽

Jacobi Weight ◽

Orthogonal Sequences

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the ℤ3-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function.

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