Cubic Decomposition of 2-Orthogonal Polynomial Sequences

Maroni, P.; Mesquita, Teresa A.

doi:10.1007/s00009-012-0223-3

Cited by 5 publications

(14 citation statements)

References 14 publications

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“…n } n 0 as the diagonal components of the cubic decomposition of the threefold symmetric sequence {P n } n 0 , which is line with the terminology adopted in a more general cubic decomposition framework in [33].…”

Section: [J]supporting

confidence: 66%

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

Loureiro

Assche

2019

Anal. Appl.

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We characterize all the multiple orthogonal threefold symmetric polynomial sequences whose sequence of derivatives is also multiple orthogonal. Such a property is commonly called the Hahn property and it is an extension of the concept of classical polynomials to the context of multiple orthogonality. The emphasis is on the polynomials whose indices lie on the step line, also known as 2-orthogonal polynomials. We explain the relation of the asymptotic behavior of the recurrence coefficients to that of the largest zero (in absolute value) of the polynomial set. We provide a full characterization of the Hahn-classical orthogonality measures supported on a 3-star in the complex plane containing all the zeros of the polynomials. There are essentially three distinct families, one of them 2-orthogonal with respect to two confluent functions of the second kind. This paper complements earlier research of Douak and Maroni.

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Section: [J]supporting

confidence: 66%

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

Loureiro

Assche

2019

Anal. Appl.

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“…. , d + 1, be the Darboux transformations of J 1 given by the d cyclid permutations of the matrices in the right hand side of (25). Let {A j n }, j = 2, .…”

Section: (D+1)n 11mentioning

confidence: 99%

“…On the other hand, following the ideas in [24] (and the references therein) for standard sequences of orthogonal polynomials, in [26] (see also [13]) the authors provide a cubic decomposition for sequences of polynomials, multiple orthogonal with respect to a two different linear functionals. Concerning the symmetric case, in [25] this cubic decomposition is analyzed for a 2-symmetric sequence of polynomials, which is called a diagonal cubic decomposition (CD) by the authors. Here, we also extend this notion of diagonal decomposition to a more general case, considering symmetric sequences of polynomials multiple orthogonal with respect to d > 3 linear functionals.…”

Section: Introductionmentioning

confidence: 99%

The symmetrization problem for multiple orthogonal polynomials

Branquinho,

Huertas

2013

Preprint

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We analyze the effect of symmetrization in the theory of multiple orthogonal polynomials. For a symmetric sequence of type II multiple orthogonal polynomials satisfying a high-term recurrence relation, we fully characterize the Weyl function associated to the corresponding block Jacobi matrix as well as the Stieltjes matrix function. Next, from an arbitrary sequence of type II multiple orthogonal polynomials with respect to a set of d linear functionals, we obtain a total of d + 1 sequences of type II multiple orthogonal polynomials, which can be used to construct a new sequence of symmetric type II multiple orthogonal polynomials. Finally, we prove a Favard-type result for certain sequences of matrix multiple orthogonal polynomials satisfying a matrix four-term recurrence relation with matrix coefficients.

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“…Therefore, we have considered wider versions of these decompositions, for which an interesting feature is obtained that both the quadratic and the cubic decomposition of the canonical sequence fulfil. Moreover, these generalized decompositions with several parameters allowed already the appearance of new examples, as for instance, in the study of the so-called diagonal cubic decomposition of 2-orthogonal polynomial sequences, [5] where the only three sequences present in the decomposition are fully *Corresponding author. Emails: teresam@portugalmail.com; teresam@portugalmail.pt characterized, together with the structure coefficients of the original (decomposed) PS which is not, in many cases, 2-symmetric.…”

Section: Introductionmentioning

confidence: 99%

“…[2,3] At the same time, and being already understood the connection between the 2-symmetric character with a simple cubic decomposition, for example , [4] a wide cubic decomposition was stated and developed. [5,6] As it is examined during this text, the simplest version of such decompositions, when applied to the sequence of monomials {x n } n≥0 , usually called canonical sequence, provides only trivial polynomial components. Therefore, we have considered wider versions of these decompositions, for which an interesting feature is obtained that both the quadratic and the cubic decomposition of the canonical sequence fulfil.…”

Section: Introductionmentioning

confidence: 99%

Chebyshev polynomials via quadratic and cubic decompositions of the canonical sequence

Mesquita

Macedo

2015

Integral Transforms and Special Functions

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In this paper we give general quadratic and cubic decompositions of the sequence of monomials {x n } n≥0 and prove how the correspondent component sequences are connected with Chebyshev polynomial sequences (PSs). The two-way street that joins together the polynomial sequences brought by quadratic or cubic decomposition and those famous PSs is established by the reversed polynomials and initial perturbations. In the cubic decomposition case, the relation is given through a 2-orthogonal Chebyshev PS.

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Cubic Decomposition of 2-Orthogonal Polynomial Sequences

Cited by 5 publications

References 14 publications

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

Three-fold symmetric Hahn-classical multiple orthogonal polynomials

The symmetrization problem for multiple orthogonal polynomials

Chebyshev polynomials via quadratic and cubic decompositions of the canonical sequence

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