A characterization of some q-orthogonal polynomials

Datta, Somjit; Griffin, James

doi:10.1007/s11139-006-0152-5

Cited by 13 publications

(7 citation statements)

References 4 publications

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“…Remark 3.5. In [5] the authors claim: "We show that the only orthogonal polynomials satisfying a q−difference equation of the form π(x)D q P n (x) = (α n x + β n )P n (x) + γ n P n−1 (x), where π(x) is a polynomial of degree 2, are the Al-Salam Carlitz 1, little and big q−Laguerre, the little and big q−Jacobi, and the q−Bessel polynomials. This is a q−analog of the work carried out in [1]."…”

Section: This Means Thatmentioning

confidence: 99%

On discrete coherent pairs of measures

Castillo¹,

Mbouna²,

Petronilho³

2020

Preprint

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In [Castillo & Mbouna, Indag. Math. 31 (2020) [223][224][225][226][227][228][229][230][231][232][233][234], the concept of π N -coherent pairs of order (m, k) with index M is introduced. This definition, implicitly related with the standard derivative operator, automatically leaves out the so-called discrete orthogonal polynomials. The purpose of this note is twofold: first we use the (discrete) Hahn difference operator and rewrite the known results in this framework; second, as an application, we describe exhaustively the (discrete) self-coherent pairs in the situation whether M = 0, N ≤ 2, and (m, k) = (1, 0). This is proved by describing in a unified way the classical orthogonal polynomials with respect to Jackson's operator as special or limiting cases of a four parametric family of q−polynomials.

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Section: This Means Thatmentioning

confidence: 99%

On discrete coherent pairs of measures

Castillo¹,

Mbouna²,

Petronilho³

2020

Preprint

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“…Since {P n } ∞ n=0 is monic and orthogonal, there exist sequences {a n } ∞ n=0 and {b n } ∞ n=1 such that the recurrence relation (1) is satisfied. If f (x) = x − a n , it follows from (7) and ( 8) that…”

Section: Preliminariesmentioning

confidence: 99%

“…Datta and Griffin [7] characterized the big q-Jacobi polynomial or one of its special or limiting cases (Al-Salam-Carlitz 1, little and big q-Laguerre, little q-Jacobi, and q-Bessel polynomials) as the only orthogonal polynomials that satisfy π(x)D q P n (x) = 1 j=−1 a n,n+j P n+j , n = 1, 2, . .…”

Section: Introductionmentioning

confidence: 99%

A characterization of Askey-Wilson polynomials

Nangho¹,

Jordaan²

2019

Proc. Amer. Math. Soc.

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We show that the only monic orthogonal polynomials {Pn} ∞ n=0 that satisfyan,n+jPn+j (x), x = cos θ, an,n−2 = 0, n = 2, 3, . . . ,where π(x) is a polynomial of degree at most 4 and Dq is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.

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“…The first structure relation, second structure relation and Bochner's theorem have been generalized to orthogonal polynomials involving the difference and q-difference operator (cf. [3,9,11,19,20]) and play an important role when studying properties of zeros or connection and linearization problems involving polynomials (see, for example, [15,19]).…”

Section: Introductionmentioning

confidence: 99%

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable

Nangho¹,

Jordaan²

2018

SIGMA

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We prove an equivalence between the existence of the first structure relation satisfied by a sequence of monic orthogonal polynomials {P n } ∞ n=0 , the orthogonality of the second derivatives {D 2x P n } ∞ n=2 and a generalized Sturm-Liouville type equation. Our treatment of the generalized Bochner theorem leads to explicit solutions of the difference equation [Vinet L., Zhedanov A., J. Comput. Appl. Math. 211 (2008), 45-56], which proves that the only monic orthogonal polynomials that satisfy the first structure relation are Wilson polynomials, continuous dual Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tend to ∞. This work extends our previous result [arXiv:1711.03349] concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying the first structure relation.

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A characterization of some q-orthogonal polynomials

Cited by 13 publications

References 4 publications

On discrete coherent pairs of measures

On discrete coherent pairs of measures

A characterization of Askey-Wilson polynomials

Structure Relations of Classical Orthogonal Polynomials in the Quadratic and q-Quadratic Variable

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