Lowering and raising operators for some special orthogonal polynomials

Koornwinder, Tom H.

doi:10.1090/conm/417/07924

Cited by 9 publications

(9 citation statements)

References 12 publications

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“…We remark again that the raising operator shouldn't be confused with the backward shift operator in [29] which changes the value of parameters of the distribution ρ.…”

Section: Definition 21 (Hypergeometric Function) the Hypergeometric mentioning

confidence: 97%

“…Remark that the raising operator increases the degree of the polynomial by one, similarly to the socalled backward shift operator [28]. However the raising operator in (24) does not change the parameters involved in the function ρ, whereas the backward operator increases the degree and lowers the parameters [29].…”

Section: Definition 21 (Hypergeometric Function) the Hypergeometric F...mentioning

confidence: 99%

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Stochastic Duality and Orthogonal Polynomials

Franceschini

Giardinà

2019

Springer Proceedings in Mathematics &Amp; Statistics

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For a series of Markov processes we prove stochastic duality relations with duality functions given by orthogonal polynomials. This means that expectations with respect to the original process (which evolves the variable of the orthogonal polynomial) can be studied via expectations with respect to the dual process (which evolves the index of the polynomial). The set of processes include interacting particle systems, such as the exclusion process, the inclusion process and independent random walkers, as well as interacting diffusions and redistribution models of Kipnis-Marchioro-Presutti type. Duality functions are given in terms of classical orthogonal polynomials, both of discrete and continuous variable, and the measure in the orthogonality relation coincides with the process stationary measure.

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“…We remark again that the raising operator shouldn't be confused with the backward shift operator in [29] which changes the value of parameters of the distribution ρ.…”

Section: Definition 21 (Hypergeometric Function) the Hypergeometric mentioning

confidence: 97%

Section: Definition 21 (Hypergeometric Function) the Hypergeometric F...mentioning

confidence: 99%

Stochastic Duality and Orthogonal Polynomials

Franceschini

Giardinà

2019

Springer Proceedings in Mathematics &Amp; Statistics

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“…In §5 this is specialized to the case of continuous q-Jacobi polynomials and we show that it has the results for Jacobi polynomials as a limit case. A further specialization to continuous q-ultraspherical polynomials is given in §6, and the resulting structure relation is related to another one obtained from results in [15]. Finally, in §7, we take the limit of the Askey-Wilson case to the case of big q-Jacobi polynomials, and we relate the resulting structure relation to the one in [18].…”

Section: Introductionmentioning

confidence: 95%

“…Variants of lowering and raising relations (1.2), (1.3) are scattered over the literature. See a brief survey in [15]. Note in particular the lowering and raising relations for A n type Macdonald polynomials given by Kirillov & Noumi [13].…”

Section: Introductionmentioning

confidence: 99%

The structure relation for Askey–Wilson polynomials

Koornwinder

2007

Journal of Computational and Applied Mathematics

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This paper is dedicated to Nico Temme on the occasion of his 65th birthday. AbstractAn explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n + 1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.

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“…This question was also raised by Tom Koornwinder at the Edinburgh conference on symmetric functions organized by Vadim Kuznetsov. The case n = 1 corresponds to the celebrated Askey-Wilson polynomials and Koornwinder's paper [14] from that conference contains partial results in this direction as well as a survey of earlier results.…”

Section: Introductionmentioning

confidence: 99%

Raising and Lowering Operators for Askey-Wilson Polynomials

Sahi¹

2007

SIGMA

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Abstract. In this paper we describe two pairs of raising/lowering operators for AskeyWilson polynomials, which result from constructions involving very different techniques. The first technique is quite elementary, and depends only on the "classical" properties of these polynomials, viz. the q-difference equation and the three term recurrence. The second technique is less elementary, and involves the one-variable version of the double affine Hecke algebra.

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Lowering and raising operators for some special orthogonal polynomials

Cited by 9 publications

References 12 publications

Stochastic Duality and Orthogonal Polynomials

Stochastic Duality and Orthogonal Polynomials

The structure relation for Askey–Wilson polynomials

Raising and Lowering Operators for Askey-Wilson Polynomials

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