New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

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

doi:10.1016/j.jmaa.2010.03.024

Cited by 4 publications

(3 citation statements)

References 20 publications

(17 reference statements)

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“…Remark 3.6. Note that if α = −2N − 1 and β ∈ C and due to (3.9) one must consider the moment linear form [10] L…”

Section: Big −1 Jacobi Polynomialsmentioning

confidence: 99%

The Orthogonality of Al-Salam–Carlitz Polynomials for Complex Parameters

Cohl¹,

Costas-Santos²,

Xu³

2018

Frontiers in Orthogonal Polynomials and q-Series

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In this contribution, we study the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials U (a) n (x; q) when the parameters a and q are not necessarily real nor 'classical', i.e., the linear functional u with respect to such polynomial sequence is quasi-definite and not positive definite. We establish orthogonality on a simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the Al-Salam-Carlitz polynomials U (a) n (x; q) of degree n up to a constant factor. We also obtain a generalization of the unique generating function for these polynomials.

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“…Remark 3.6. Note that if α = −2N − 1 and β ∈ C and due to (3.9) one must consider the moment linear form [10] L…”

Section: Big −1 Jacobi Polynomialsmentioning

confidence: 99%

The Orthogonality of Al-Salam–Carlitz Polynomials for Complex Parameters

Cohl¹,

Costas-Santos²,

Xu³

2018

Frontiers in Orthogonal Polynomials and q-Series

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“…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%

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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Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

Moreno¹,

García−Caballero²

2011

Taiwanese J. Math.

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We consider the continuous q-Jacobi polynomials {Pn=0 , extending the variable and the parameters beyond classical considerations. For those new allowed values of the parameters for which Favard's theorem fails to work, we construct inner products in which the presence of the Askey-Wilson divided difference operator provides the q-Sobolev character of the non-standard orthogonality for the corresponding family.

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New orthogonality relations for the continuous and the discrete q-ultraspherical polynomials

Cited by 4 publications

References 20 publications

The Orthogonality of Al-Salam–Carlitz Polynomials for Complex Parameters

The Orthogonality of Al-Salam–Carlitz Polynomials for Complex Parameters

Orthogonality of q-polynomials for non-standard parameters

Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

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