Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

Moreno, Samuel G.; García−Caballero, Esther M.

doi:10.11650/twjm/1500406372

Cited by 3 publications

(5 citation statements)

References 18 publications

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“…A slightly less detailed study of orthogonality conditions for the big q-Jacobi polynomials can be found in [17], using a bilinear form instead of the linear form L 0 .…”

Section: The Big Q-jacobi Polynomialsmentioning

confidence: 99%

“…the operator T in Theorem 2.2 can be chosen as D q −1 , and condition (7) holds (see the relation between q-Hahn and big q-Jacobi polynomials and the expression (19) for the coefficients γ n ). Also, for n ≥ N ,…”

Section: The Orthogonality Conditions For |Q| <mentioning

confidence: 99%

“…which can be obtained easily from the hypergeometric representation (17) or from the TTRR (18), this case is reduced to that in Section 4.1.1. Multiplying relation (20) by the factor (b − q −N ), taking the limit b → q −N , and removing some non-vanishing constants, one gets, for n ̸ = m,…”

Section: The Orthogonality Conditions For |Q| <mentioning

confidence: 99%

“…The second term is relevant for polynomials with degree greater than N , and it is needed in order to have an orthogonality characterizing all the sequence ( p n ) ∞ n=0 . The technique used in [3] is applicable for all classical orthogonal polynomials with one coefficient γ n vanishing, and in fact it has been used recently in [16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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Orthogonality of q-polynomials for non-standard parameters

Costas-Santos

Sánchez-Lara

2011

Journal of Approximation Theory

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q-polynomials can be defined for all the possible parameters, but their orthogonality properties are unknown for several configurations of the parameters. Indeed, orthogonality for the Askey-Wilson polynomials, p n (x; a, b, c, d; q), is known only when the product of any two parameters a, b, c, d is not a negative integer power of q. Also, the orthogonality of the big q-Jacobi, p n (x; a, b, c; q), is known when a, b, c, abc −1 is not a negative integer power of q. In this paper, we obtain orthogonality properties for the Askey-Wilson polynomials and the big q-Jacobi polynomials for the rest of the parameters and for all n ∈ N 0 . For a few values of such parameters, the three-term recurrence relation (TTRR)x p n = p n+1 + β n p n + γ n p n−1 , n ≥ 0, presents some index for which the coefficient γ n = 0, and hence Favard's theorem cannot be applied. For this purpose, we state a degenerate version of Favard's theorem, which is valid for all sequences of polynomials satisfying a TTRR even when some coefficient γ n vanishes, i.e., {n : γ n = 0} ̸ = ∅.We also apply this result to the continuous dual q-Hahn, big q-Laguerre, q-Meixner, and little qJacobi polynomials, although it is also applicable to any family of orthogonal polynomials, in particular the classical orthogonal polynomials.

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“…A slightly less detailed study of orthogonality conditions for the big q-Jacobi polynomials can be found in [17], using a bilinear form instead of the linear form L 0 .…”

Section: The Big Q-jacobi Polynomialsmentioning

confidence: 99%

Section: The Orthogonality Conditions For |Q| <mentioning

confidence: 99%

Section: The Orthogonality Conditions For |Q| <mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Orthogonality of q-polynomials for non-standard parameters

Costas-Santos

Sánchez-Lara

2011

Journal of Approximation Theory

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show abstract

“…A slightly less detailed study on orthogonality conditions for the big q-Jacobi can be found in [27].…”

Section: The Big Q-jacobi Polynomialsmentioning

confidence: 99%

A Survey on q-Polynomials and their Orthogonality Properties

Costas-Santos,

Sanchez-Lara

2010

Preprint

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In this paper we study the orthogonality conditions satisfied by the classical q-orthogonal polynomials that are located at the top of the q-Hahn tableau (big q-jacobi polynomials (bqJ)) and the Nikiforov-Uvarov tableau (Askey-Wilson polynomials (AW)) for almost any complex value of the parameters and for all non-negative integers degrees.We state the degenerate version of Favard's theorem, which is one of the keys of the paper, that allow us to extend the orthogonality properties valid up to some integer degree N to Sobolev type orthogonality properties.We also present, following an analogous process that applied in [16], tables with the factorization and the discrete Sobolev-type orthogonality property for those families which satisfy a finite orthogonality property, i.e. it consists in sum of finite number of masspoints, such as q-Racah (qR), q-Hahn (qH), dual q-Hahn (dqH), and q-Krawtchouk polynomials (qK), among others.

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On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

Sánchez-Lara¹

2017

Rocky Mountain J. Math.

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Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

Cited by 3 publications

References 18 publications

Orthogonality of q-polynomials for non-standard parameters

Orthogonality of q-polynomials for non-standard parameters

A Survey on q-Polynomials and their Orthogonality Properties

On the Sobolev orthogonality of classical orthogonal polynomials for non standard parameters

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