The combinatorics of Al-Salam–Chihara q-Laguerre polynomials

Kasraoui, Anisse; Stanton, Dennis; Zeng, Jiang

doi:10.1016/j.aam.2010.04.008

Cited by 32 publications

(28 citation statements)

References 18 publications

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“…Since then, combinatorial formulas have been given for the moments of (the weight functions of) many of the polynomials in the Askey scheme, including q-Hermite, Chebyshev, q-Laguerre, Charlier, Meixner, and Al-Salam-Chihara polynomials, see e.g. [27,28,30,34,41]. However, no such formula was known for the moments of the Askey-Wilson polynomials.…”

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confidence: 99%

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Corteel

Williams²

2011

Duke Math. J.

178

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Abstract. Introduced in the late 1960's [33,43], the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It has been cited as a model for traffic flow and protein synthesis. In the most general form of the ASEP with open boundaries, particles may enter and exit at the left with probabilities α and γ, and they may exit and enter at the right with probabilities β and δ. In the bulk, the probability of hopping left is q times the probability of hopping right. The first main result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of a new class of tableaux which we call staircase tableaux. This generalizes our previous work [14,15] for the ASEP with parameters γ = δ = 0. Using our first result and also results of Uchiyama-Sasamoto-Wadati [48], we derive our second main result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials (e.g. Hermite, Charlier, Laguerre, Meixner). However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.

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Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Corteel

Williams²

2011

Duke Math. J.

178

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“…They appear in the combinatorial interpretation of the moments of Al-Salam-Chihara q-Laguerre polynomials [KSZ08]. These are another q-analogue of Laguerre polynomials whose recurrence relation is:…”

Section: Permutations and Crossingsmentioning

confidence: 99%

The Matrix Ansatz, orthogonal polynomials, and permutations

Corteel¹,

Josuat-Vergès²,

Williams³

2011

Advances in Applied Mathematics

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In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We illustrate this approach with applications to moments of orthogonal polynomials, permutations, signed permutations, and tableaux.To Dennis Stanton with admiration.where Z n = W |(D + E) n |V .1991 Mathematics Subject Classification. 05A15,05A19,33C45.

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“…The other specialisation was introduced by Kasraoui, Stanton and Zeng [13], however, without providing a formula for the moments (these are actually a particular case of octabasic q-Laguerre polynomials from [23]). Theorem 2.5 (Kasraoui, Stanton, Zeng [13]). Define q-Laguerre polynomialsL n = L n (x; y|q) as…”

Section: Particular Classes Of Orthogonal Polynomialsmentioning

confidence: 99%

Crossings, Motzkin paths and moments

Josuat-Vergès

Rubey

2011

Discrete Mathematics

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Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain q-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the q-Laguerre and the q-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some q-Hermite polynomials, and also the distribution of crossings in matchings).Our method mainly consists in the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard-Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.

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The combinatorics of Al-Salam–Chihara q-Laguerre polynomials

Cited by 32 publications

References 18 publications

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials

The Matrix Ansatz, orthogonal polynomials, and permutations

Crossings, Motzkin paths and moments

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