Exceptional Meixner and Laguerre orthogonal polynomials

Durán, Antonio J.

doi:10.1016/j.jat.2014.05.009

Cited by 63 publications

(138 citation statements)

References 41 publications

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“…We fix the quotient and derive the asymptotic behavior of the remainder polynomial

R_{λ}

when the size of the core

k (k + 1) / 2

tends to infinity; see Theorem . Section 6. We identify Wronskian Hermite polynomials with Wronskians involving Laguerre polynomials . The obtained identity is naturally labeled by the partition and its core and quotient; see Proposition .…”

Section: Introductionmentioning

confidence: 99%

Coefficients of Wronskian Hermite polynomials

2020

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We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the p‐star.

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“…We fix the quotient and derive the asymptotic behavior of the remainder polynomial

R_{λ}

when the size of the core

k (k + 1) / 2

Section: Introductionmentioning

confidence: 99%

Coefficients of Wronskian Hermite polynomials

2020

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“…These polynomials are defined just like classical orthogonal polynomials, albeit with the possibility that for a finite number of degrees no polynomial exists. In the primary examples, these exceptional orthogonal polynomials appear as Wronskians of a set of classical orthogonal polynomials [29], and as such are called exceptional Hermite [11,17], exceptional Laguerre [5,12] and exceptional Jacobi [14] polynomials.…”

Section: Introductionmentioning

confidence: 99%

Recurrence Relations for Wronskian Hermite Polynomials

Bonneux¹,

Stevens²

2018

SIGMA

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Abstract. We consider polynomials that are defined as Wronskians of certain sets of Hermite polynomials. Our main result is a recurrence relation for these polynomials in terms of those of one or two degrees smaller, which generalizes the well-known three term recurrence relation for Hermite polynomials. The polynomials are defined using partitions of natural numbers, and the coefficients in the recurrence relation can be expressed in terms of the number of standard Young tableaux of these partitions. Using the recurrence relation, we provide another recurrence relation and show that the average of the considered polynomials with respect to the Plancherel measure is very simple. Furthermore, we show that some existing results in the literature are easy corollaries of the recurrence relation.

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“…Durán [16] reported multi-indexed Meixner polynomials, in which he used both eigenvectors and virtual state vectors as seed solutions. The method of construction consists in dualizing Krall discrete orthogonal polynomials.…”

Section: Summary and Commentsmentioning

confidence: 99%

“…Meixner case was studied by Durán [16]. In our language his polynomials correspond to the eigenstates and/or virtual states deletion.…”

Section: Introductionmentioning

confidence: 99%

Multi-indexed Meixner and littleq-Jacobi (Laguerre) polynomials

Одаке¹,

Sasaki²

2017

J. Phys. A: Math. Theor.

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As the fourth stage of the project multi-indexed orthogonal polynomials, we present the multi-indexed Meixner and little q-Jacobi (Laguerre) polynomials in the framework of 'discrete quantum mechanics' with real shifts defined on the semi-infinite lattice in one dimension. They are obtained, in a similar way to the multi-indexed Laguerre and Jacobi polynomials reported earlier, from the quantum mechanical systems corresponding to the original orthogonal polynomials by multiple application of the discrete analogue of the Darboux transformations or the Crum-Krein-Adler deletion of virtual state vectors. The virtual state vectors are the solutions of the matrix Schrödinger equation on all the lattice points having negative energies and infinite norm. This is in good contrast to the (q-)Racah systems defined on a finite lattice, in which the 'virtual state' vectors satisfy the matrix Schrödinger equation except for one of the two boundary points.

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Exceptional Meixner and Laguerre orthogonal polynomials

Cited by 63 publications

References 41 publications

Coefficients of Wronskian Hermite polynomials

Coefficients of Wronskian Hermite polynomials

Recurrence Relations for Wronskian Hermite Polynomials

Multi-indexed Meixner and littleq-Jacobi (Laguerre) polynomials

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