Sobolev orthogonality for the Gegenbauer polynomials {Cn(−N+12)}n⩾0

Morales, María Álvarez de; Pérez, Teresa E.; Piñar, Miguel A.

doi:10.1016/s0377-0427(98)00124-1

Cited by 30 publications

(27 citation statements)

References 2 publications

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“…In this respect, we refer the reader to [2][3][4]11,12] where general results on the Sobolev orthogonality of the Jacobi or Gegenbauer polynomials when one or both parameters α and β are negative integers. (…”

Section: Introductionmentioning

confidence: 99%

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Bruder

Littlejohn

2011

Results. Math.

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In this paper, we consider the second-order Jacobi differential expressionhere, the Jacobi parameters are α > −1 and β = −1. This is a nonclassical setting since the classical setting for this expression is generally considered when α, β > −1. In the classical setting, it is well-known that the Jacobi polynomials {P (α,β) n } ∞ n=0 are (orthogonal) eigenfunctions of a self-adjoint operator T α,β , generated by the Jacobi differential expression, in the Hilbert space L 2 ((−1, 1); (1 − x) α (1 + x) β ). When α > −1 and β = −1, the Jacobi polynomial of degree 0 does not belong to the Hilbert space L 2 ((−1, 1); (1 − x) α (1 + x) −1 ). However, in this paper, we show that the full sequence of Jacobi polynomials {P (α,−1) n } ∞ n=0 forms a complete orthogonal set in a Hilbert-Sobolev space Wα, generated by the inner productWe also construct a self-adjoint operator Tα, generated by α,−1 [·] in Wα, that has the Jacobi polynomials {P (α,−1) n } ∞ n=0 as eigenfunctions. Mathematics Subject Classification (2010). Primary 33C45, 34B30, 47B25; Secondary 34B20, 47B65. 284 A. Bruder and L. L. Littlejohn Results. Math.

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Section: Introductionmentioning

confidence: 99%

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Bruder

Littlejohn

2011

Results. Math.

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“…In 1998,Álvarez de Morales et al [3] found, using a different technique, the orthogonality for the Gegenbauer polynomials C (−N +1/2) n . The case of Jacobi polynomials with negative integer parameters was studied by Pérez and Piñar et al [1,2].…”

Section: Introductionmentioning

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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“…In the case of d = 1, the equation (1.1) becomes the ordinary differential equation satisfied by the Gegenbauer polynomials. In this case, the problem of negative indices has been studied by several authors; we refer to [1,2,3,6] and the references therein. For d = 2, equation (1.1) is classical and can be traced back to Hermite.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal polynomials and partial differential equations on the unit ball

Piñar¹,

Xu²

2009

Proc. Amer. Math. Soc.

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Abstract. Orthogonal polynomials of degree n with respect to the weight function W µ (x) = (1 − x 2 ) µ on the unit ball in R are known to satisfy the partial differential equationThe singular case of µ = −1, −2, . . . is studied in this paper. Explicit polynomial solutions are constructed and the equation for ν = −2, −3, . . . is shown to have complete polynomial solutions if the dimension d is odd. The orthogonality of the solution is also discussed.

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Sobolev orthogonality for the Gegenbauer polynomials {Cn(−N+12)}n⩾0

Cited by 30 publications

References 2 publications

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

Orthogonality of q-polynomials for non-standard parameters

Orthogonal polynomials and partial differential equations on the unit ball

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