Non-classical orthogonality relations for big and little <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>q</mml:mi></mml:math>-Jacobi polynomials

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

doi:10.1016/j.jat.2009.05.008

Journal of Approximation Theory

2010

DOI: 10.1016/j.jat.2009.05.008

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Non-classical orthogonality relations for big and little q-Jacobi polynomials

Samuel G. Moreno

Esther M. García−Caballero

Abstract: Big q-Jacobi polynomials {P n (·; a, b, c; q)} ∞ n=0 are classically defined for 0 < a < q −1 , 0 < b < q −1 and c < 0. For the family of little q-Jacobi polynomials { p n (·; a, b|q)} ∞ n=0 , classical considerations restrict the parameters imposing 0 < a < q −1 and b < q −1 . In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditions for those cases for which Favard's theorem does not work. As a by-product, we obtain similar results… Show more

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Cited by 5 publications

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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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Orthogonality of q-polynomials for non-standard parameters

Costas-Santos

Sánchez-Lara

2011

Journal of Approximation Theory

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

Moreno

García−Caballero

2010

Journal of Mathematical Analysis and Applications

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Keywords:Continuous q-ultraspherical polynomials Discrete q-ultraspherical polynomials Non-standard orthogonality Favard's theorem In a recent contribution [N.M. Atakishiyev, A.U. Klimyk, On discrete q-ultraspherical polynomials and their duals, J. Math. Anal. Appl. 306 (2005) 637-645], the so-named discrete q-ultraspherical polynomials were introduced as a specialization of the big qJacobi polynomials, and their orthogonality established for values of the parameter outside its commonly known domain but inside the range of validity of the conditions of Favard's theorem. In this paper we consider both the continuous and the discrete q-ultraspherical polynomials and we prove that their orthogonality is guaranteed for the whole range of the allowed parameters, even in those intriguing cases in which the three term recurrence relation breaks down. The presence of either the Askey-Wilson divided difference operator (in the continuous case), or the q-derivative operator (in the discrete one), provides the q-Sobolev character of the non-standard inner products introduced in our approach.

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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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Non-classical orthogonality relations for big and little q-Jacobi polynomials

Cited by 5 publications

References 15 publications

Orthogonality of q-polynomials for non-standard parameters

Orthogonality of q-polynomials for non-standard parameters

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

Non-classical Orthogonality Relations for Continuous q-Jacobi Polynomials

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