Complex Exceptional Orthogonal Polynomials and Quasi-invariance

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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“…In the context of orthogonality for exceptional Hermite polynomials, one is interested in even partitions Lemma An even partition

λ = (λ_{1}^{2}, λ_{2}^{2}, \dots, λ_{l}^{2})

has empty core and quotient given by

false(μ, ν false) = (false(⌈ λ_{1} / 2 ⌉, ⌈ λ_{2} / 2 ⌉, \dots, ⌈ λ_{l} / 2 ⌉ false), false(⌊ λ_{1} / 2 ⌋, ⌊ λ_{2} / 2 ⌋, \dots, ⌊ λ_{l} / 2 ⌋ false)) .

…”

Section: Preliminariesmentioning

confidence: 99%

“…These Wronskian Hermite polynomials appear in the theory of exceptional orthogonal polynomials and the related topic of rational extensions of the quantum harmonic oscillator . These polynomials are well studied.…”

Section: Introductionmentioning

confidence: 99%

Coefficients of Wronskian Hermite polynomials

2020

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“…Exceptional Hermite polynomials [11,17,21] appeared as a class of exceptional orthogonal polynomials. They are defined as…”

Section: Exceptional Hermite Polynomialsmentioning

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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“…In the context of classical orthogonal polynomials, Darboux transformations are essentially a factorization of the differential operator, and lead directly to exceptional orthogonal polynomials, a new class of Sturm-Liouville polynomial families originally introduced in [4,5], and developed over the past decade by many authors [6][7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials

Gómez‐Ullate¹

2018

J. Phys. A: Math. Theor.

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In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Pöschl-Teller potential.2010 Mathematics Subject Classification. 33C45, 81Q80, 42C05.

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Complex Exceptional Orthogonal Polynomials and Quasi-invariance

Cited by 5 publications

References 18 publications

Coefficients of Wronskian Hermite polynomials

Coefficients of Wronskian Hermite polynomials

Recurrence Relations for Wronskian Hermite Polynomials

Shape invariance and equivalence relations for pseudo-Wronskians of Laguerre and Jacobi polynomials

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