A non-symmetric second-degree semi-classical form of class one

Alaya, A.; Bouras, B.; Marcellán, Francisco

doi:10.1080/10652469.2011.567388

Cited by 9 publications

(3 citation statements)

References 13 publications

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“…First family: φ(x) = x x 3 + τ 3 8 , ψ(x) = −3(p + q + 1)x 3 + τ x 2 − 1 2 τ 2 x − 1 8 τ 3 3q − 1 2 , (w) 0 = 1, (w) 1 = τ, (w) 2 = τ 2 4 , τ = 0, p + q ≥ 0, p, q ∈ Z. Second family: φ(x) = x x 3 + τ 3 , ψ(x) = −3(p + q + 1)x 3 + τ x 2 − τ 2 x − τ 3 3q + 1 2 , (w) 0 = 1, (w) 1 = τ, (w) 2 = 0, τ = 0, p + q ≥ 0, p, q ∈ Z. φ(x) = x(x 2 − 1), ψ(x) = −2(k + l + 2)x 2 + x + 2l − 2q 3 + 1, k + l ≥ −1, k, l ∈ Z, q ∈ {1, 2}. CD: W 3n (x) = P n x 3 + q ′ x + r , q ′ , r ∈ C. Second family: φ(x) = x x 3 + τ 3 8 , ψ(x) = −3(k + l + 2)x 3 + τ x 2 − 1 2 τ 2 x − 1 8 τ 3 (3l − q + 1), (w) 0 = 1, (w) 1 = τ, (w) 2 = τ 2 4 , τ = 0, k + l ≥ −1, k, l ∈ Z, q ∈ {1, 2}.…”

Section: Description Via Third Degree Character Of a Family Of Semicl...mentioning

confidence: 99%

Several characterizations of third degree semiclassical linear forms of class two appearing via cubic decomposition

Khalfallah

2022

Preprint

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In this work we consider orthogonal polynomials via cubic decomposition in the framework of the third degree semiclassical class. Based on their third degree character, we give a complete description, by using the formal Stieltjes function and the moments, of semiclassical linear forms of class two arising from the cubic decomposition W3n(x) = Pn(x3); n ≥ 0. We focus our attention on the link between these forms and the strict third degree classical forms Vqk,l= J (k + q/3; l -q/3), q ∈{1,2},k,l ∈Z with k+ l≥ -1. All of them are rational transformations of the Jacobi form V = J (-2/3;-1=3).

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Section: Description Via Third Degree Character Of a Family Of Semicl...mentioning

confidence: 99%

Several characterizations of third degree semiclassical linear forms of class two appearing via cubic decomposition

Khalfallah

2022

Preprint

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“…Stieltjes transforms satisfying quadratic equations with polynomial coefficients are called second degree forms; see [2], [6], [45], and [49].…”

Section: Orthogonal Polynomialsmentioning

confidence: 99%

Orthogonality of the Dickson polynomials of the $(k+1)$-th kind

Dominici¹

2021

Rev. Un. Mat. Argentina

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“…Stieltjes transforms satisfying quadratic equations with polynomial coefficients are called second degree forms. See [2], [6], [43], and [48].…”

Section: Orthogonal Polynomialsmentioning

confidence: 99%

Orthogonality of the Dickson polynomials of the (k+1)-th kind

Dominici¹

2020

Preprint

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We study the Dickson polynomials of the (k + 1)-th kind over the field of complex numbers. We show that they are a family of co-recursive orthogonal polynomials with respect to a quasi-definite moment functional L k . We find an integral representation for L k and compute explicit expressions for all of its moments. Dedicated to V. E. P., in gratitude for her continuous unbounded support (without measure).

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A non-symmetric second-degree semi-classical form of class one

Cited by 9 publications

References 13 publications

Several characterizations of third degree semiclassical linear forms of class two appearing via cubic decomposition

Several characterizations of third degree semiclassical linear forms of class two appearing via cubic decomposition

Orthogonality of the Dickson polynomials of the $(k+1)$-th kind

Orthogonality of the Dickson polynomials of the (k+1)-th kind

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