Mixed recurrence equations and interlacing properties for zeros of sequences of classical  q -orthogonal polynomials

Tcheutia, Daniel Duviol; Jooste, Alta; Koepf, Wolfram

doi:10.1016/j.apnum.2017.11.003

Cited by 7 publications

(3 citation statements)

References 17 publications

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“…Levit [34] was the first to study the separation of the zeros of different sequences of Hahn polynomials and interlacing results for Jacobi polynomials [4,13], Krawtchouk polynomials [9,25] and Meixner and Meixner-Pollaczek polynomials [25] followed. Interlacing results for the zeros of different sequences of q-orthogonal sequences with shifted parameters are given in [19,26,35,40]. Completed Stieltjes interlacing of zeros of different orthogonal sequences was done for the Gegenbauer [14], Laguerre [16] and Jacobi polynomials [15] and apart from the papers cited in the previous paragraph, inner bounds for the extreme zeros of Gegenbauer, Laguerre and Jacobi polynomials were also given in [2,6,21,32,36,39]; bounds for the extreme zeros of the discrete orthogonal Charlier, Meixner, Krawtchouk and Hahn polynomials in [3,33], for the extreme zeros of the q-Jacobi and q-Laguerre polynomials in [21] and for the little q-Jacobi polynomials in [19].…”

Section: Introductionmentioning

confidence: 99%

Recurrence equations involving different orthogonal polynomial sequences and applications

Jooste¹,

Tcheutia²,

Koepf³

2021

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Consider {p n } ∞ n=0 , a sequence of polynomials orthogonal with respect to w(x) > 0 on (a, b), and polynomials {g n,k } ∞ n=0 , k ∈ N 0 , orthogonal with respect to c k (x)w(x) > 0 on (a, b), where c k (x) is a polynomial of degree k in x. We show how Christoffel's formula can be used to obtain mixed three-term recurrence equations involving the polynomials p n , p n−1 and g n−m,k , m ∈ {2, 3, . . . , n−1}. In order for the zeros of p n and G m−1 g n−m,k to interlace (assuming p n and g n−m,k are co-prime), the coefficient of p n−1 , namely G m−1 , should be of exact degree m − 1, in which case restrictions on the parameter k are necessary. The zeros of G m−1 can be considered to be inner bounds for the extreme zeros of the (classical or q-classical) orthogonal polynomial p n and we give examples to illustrate the accuracy of these bounds. Because of the complexity the mixed three-term recurrence equations in each case, algorithmic tools, mainly Zeilberger's algorithm and its q-analogue, are used to obtain them.

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Section: Introductionmentioning

confidence: 99%

Recurrence equations involving different orthogonal polynomial sequences and applications

Jooste¹,

Tcheutia²,

Koepf³

2021

Preprint

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“…Our algorithms and all the equations used in this paper can be downloaded from http: //www.mathematik.uni-kassel.de/ ~koepf/Publikationen. In [27], using the same algorithmic approach, the authors provide a procedure to find mixed recurrence equations satisfied by classical q-orthogonal polynomials with shifted parameters. These equations were used to investigate interlacing properties of zeros of sequences of q-orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

Tcheutia¹,

Jooste²,

Koepf³

2018

SIGMA

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We show how to obtain linear combinations of polynomials in an orthogonal sequence {P n } n≥0 , such as Q n,k (x) = k i=0 a n,i P n−i (x), a n,0 a n,k = 0, that characterize quasiorthogonal polynomials of order k ≤ n − 1. The polynomials in the sequence {Q n,k } n≥0 are obtained from P n , by making use of parameter shifts. We use an algorithmic approach to find these linear combinations for each family applicable and these equations are used to prove quasi-orthogonality of order k. We also determine the location of the extreme zeros of the quasi-orthogonal polynomials with respect to the end points of the interval of orthogonality of the sequence {P n } n≥0 , where possible.

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“…[27]). The Askey-Wilson polynomials satisfy the contiguous relationsP n (x; a, b, c, d | q) = P n (x; aq, b, c, d | q) − C n (a, b, c, d; q) 2 P n−1 (x; aq, b, c, d | q), P n (x; aq, b, c, d | q) = P n (x; aq, bq, c, d | q) − C n (b, aq, c, d) 2 P n−1 (x; aq, bq, c, d | q), where C n (a, b, c, d) = a 1 − q n 1 − bcq n−1 1 − bdq n−1 1 − dcq n−1 1 − abcdq 2n−2 1 − abcdq 2n−1is the coefficient C n appearing in the three-term recurrence relation[18, equation (14.1.5)].…”

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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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Mixed recurrence equations and interlacing properties for zeros of sequences of classical q -orthogonal polynomials

Cited by 7 publications

References 17 publications

Recurrence equations involving different orthogonal polynomial sequences and applications

Recurrence equations involving different orthogonal polynomial sequences and applications

Quasi-Orthogonality of Some Hypergeometric and q-Hypergeometric Polynomials

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

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