Recurrence relations for exceptional Hermite polynomials

Gómez‐Ullate, David; Kasman, Alex; Kuijlaars, Arno B. J.; Milson, Robert

doi:10.1016/j.jat.2015.12.003

Cited by 52 publications

(85 citation statements)

References 17 publications

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“…2. Therefore, He can be decomposed as in (13). By evaluating the total degree, we need to have | | = |̄| + 2 deg( ), and so by (10), we have deg( ) = | | + | |.…”

Section: Factorization Of Wronskian Hermite Polynomialsmentioning

confidence: 99%

“…In this section, we obtain expressions for all coefficients of the Wronskian Hermite polynomials in terms of partition data. We consider the factorization of He given in (13) and write the remainder polynomial as…”

Section: Coefficients Of Wronskian Hermite Polynomialsmentioning

confidence: 99%

“…[8][9][10][11] These polynomials are well studied. For example, they fulfill recurrence relations, 12,13 the asymptotic behavior of their zeros is derived, 14 and moreover, via iterating rational Darboux transformations applied to the harmonic oscillator, one obtains appealing identities between (pseudo-)Wronskians involving Hermite polynomials. 15,16 Another place where they appear is in the study of rational solutions of the fourth Painlevé equation, and in that setting, they are called the generalized Hermite and generalized Okamoto polynomials, [17][18][19][20][21][22][23][24][25][26] which also appear in the rational solutions of the Boussinesq equation via a symmetry reduction to Painlevé IV.…”

Section: Introductionmentioning

confidence: 99%

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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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“…2. Therefore, He can be decomposed as in (13). By evaluating the total degree, we need to have | | = |̄| + 2 deg( ), and so by (10), we have deg( ) = | | + | |.…”

Section: Factorization Of Wronskian Hermite Polynomialsmentioning

confidence: 99%

Section: Coefficients Of Wronskian Hermite Polynomialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Coefficients of Wronskian Hermite polynomials

2020

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“…Let

λ = (λ_{1}, \dots, λ_{ℓ})

be a partition and

M \subset Z

the corresponding standard Maya diagram with

M_{+} = false{m_{1}, \dots, m_{ℓ} false}

its positive elements as determined by . Following , we define an infinite number of polynomials in the following manner:

H_{n}^{(λ)} = Wr [H_{m_{ℓ}}, \dots, H_{m_{1}}, H_{ℓ - false| λ false| + n}], n \notin M + false| λ false| - ℓ .

By construction,

H_{n}^{false(λ false)}

is a polynomial with

0true prefixdeg H_{n}^{false(λ false)} = \sum_{i = 1}^{ℓ} (() m_{i} - i + 1) + ℓ - | λ | + n - ℓ = n

The degree sequence for the exceptional Hermite family indexed by partition λ is

Z ∖ (M + | λ | - ℓ)

. Thus, degrees

0, 1, \dots, | λ | - ℓ - 1

and the degrees

m_{1} + false| λ false| - ℓ, \dots, m_{ℓ} + false| λ false| - ℓ

are missing, so that the polynomial sequence

{false{H_{n}^{false(λ false)} false}}_{n}

is missing a total of

| λ | <...>

…”

Section: Application To Exceptional Hermite Polynomialsmentioning

confidence: 99%

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

2018

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We derive identities between determinants whose entries are Hermite polynomials. These identities have a combinatorial interpretation in terms of Maya diagrams, partitions and Durfee rectangles, and serve to characterize an equivalence class of rational Darboux transformations. Since the determinants have different orders, we analyze the problem of finding the minimal order determinant in each equivalence class, and describe the solution using an elegant graphical interpretation. The results are applied to provide a more efficient representation for exceptional Hermite polynomials and for rational solutions of the Painlevé IV equation. The latter are expressed in terms of the Okamoto and generalized Hermite polynomials.

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“…Several papers are published concerning new recurrence relations for exceptional polynomials but none of these give general explicit formulas for the coefficients in the relation. Most work was done for exceptional Hermite polynomials, see for example [20] and the references therein.…”

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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Recurrence relations for exceptional Hermite polynomials

Cited by 52 publications

References 17 publications

Coefficients of Wronskian Hermite polynomials

Coefficients of Wronskian Hermite polynomials

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

Recurrence Relations for Wronskian Hermite Polynomials

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