Wronskian Appell polynomials and symmetric functions

Bonneux, Niels; Hamaker, Zachary; Stembridge, John R.; Stevens, Marco

doi:10.1016/j.aam.2019.101932

Cited by 10 publications

(39 citation statements)

References 32 publications

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“…This result follows from interpreting the generating recurrence from Ref. (section 7.2) in terms of cores and quotients. In particular, this implies that one has a similar factorization as that given in Theorem .…”

Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 87%

“…Remark In Ref. , it was shown that the average of the Wronskian polynomials with respect to the Plancherel measure is simply the monomial; that is,

\sum_{λ ⊢ n} \frac{F_{λ}^{2}}{n!} q_{λ} false(x false) = x^{n} .

As in Remark , this result is also equivalent to the orthogonality of characters, but now those evaluated in cycle types

(p^{j}, 1^{n - p j})

.…”

Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 88%

“…However, we know that for all k , the coefficient

r_{normalΦ false(μ, ν, k false), j}

is an integer by Ref. (corollary 7.1). By Lemma , we therefore conclude that

r_{normalΦ false(μ, ν, k false), j}

is, in fact, a polynomial in k of degree at most j .■…”

Section: Coefficients Of Wronskian Hermite Polynomialsmentioning

confidence: 99%

“…In that context, these polynomial sequences were studied in Ref. (section 7.2). In particular, the polynomials

q_{λ} : = \frac{Wr [q_{n_{1}}, q_{n_{2}}, \dots, q_{n_{ℓ false(λ false)}}]}{Δ (n_{λ})}

were of interest, analogous to the Wronskian Hermite polynomials .…”

Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 99%

“…The first one uses the fact that

q_{λ} false(x false) \in Z false[x false]

for any partition λ, while the other one is based on the identity

q_{λ} false(x false) = ω_{p}^{| λ |} q_{λ^{'}} false(ω_{p}^{- 1} x false)

for

ω_{p} = - \exp false(i π / p false)

; see Ref. (section 7.2). Corollary For any integer

p \geq 2

and for any partition λ with p‐core

\overset{λ}{true}

, we have

\frac{H_{non- p-fold} false(λ false)}{H (\bar{λ})} \in Z .

…”

Section: Generalization To P‐cores and P‐quotientsmentioning

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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Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 87%

“…Remark In Ref. , it was shown that the average of the Wronskian polynomials with respect to the Plancherel measure is simply the monomial; that is,

\sum_{λ ⊢ n} \frac{F_{λ}^{2}}{n!} q_{λ} false(x false) = x^{n} .

As in Remark , this result is also equivalent to the orthogonality of characters, but now those evaluated in cycle types

(p^{j}, 1^{n - p j})

.…”

Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 88%

“…However, we know that for all k , the coefficient

r_{normalΦ false(μ, ν, k false), j}

is an integer by Ref. (corollary 7.1). By Lemma , we therefore conclude that

r_{normalΦ false(μ, ν, k false), j}

is, in fact, a polynomial in k of degree at most j .■…”

Section: Coefficients Of Wronskian Hermite Polynomialsmentioning

confidence: 99%

“…In that context, these polynomial sequences were studied in Ref. (section 7.2). In particular, the polynomials

q_{λ} : = \frac{Wr [q_{n_{1}}, q_{n_{2}}, \dots, q_{n_{ℓ false(λ false)}}]}{Δ (n_{λ})}

were of interest, analogous to the Wronskian Hermite polynomials .…”

Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 99%

“…The first one uses the fact that

q_{λ} false(x false) \in Z false[x false]

for any partition λ, while the other one is based on the identity

q_{λ} false(x false) = ω_{p}^{| λ |} q_{λ^{'}} false(ω_{p}^{- 1} x false)

for

ω_{p} = - \exp false(i π / p false)

; see Ref. (section 7.2). Corollary For any integer

p \geq 2

and for any partition λ with p‐core

\overset{λ}{true}

, we have

\frac{H_{non- p-fold} false(λ false)}{H (\bar{λ})} \in Z .

…”

Section: Generalization To P‐cores and P‐quotientsmentioning

confidence: 99%

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Coefficients of Wronskian Hermite polynomials

2020

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Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

Feigin

Hallnäs

Веселов

2021

Commun. Math. Phys.

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Lassalle and Nekrasov discovered in the 1990s a surprising correspondence between the rational Calogero–Moser system with a harmonic term and its trigonometric version. We present a conceptual explanation of this correspondence using the rational Cherednik algebra and establish its quasi-invariant extension. More specifically, we consider configurations $${\mathcal {A}}$$ A of real hyperplanes with multiplicities admitting the rational Baker–Akhiezer function and use this to introduce a new class of non-symmetric polynomials, which we call $${\mathcal {A}}$$ A -Hermite polynomials. These polynomials form a linear basis in the space of $${\mathcal {A}}$$ A -quasi-invariants, which is an eigenbasis for the corresponding generalised rational Calogero–Moser operator with harmonic term. In the case of the Coxeter configuration of type $$A_N$$ A N this leads to a quasi-invariant version of the Lassalle–Nekrasov correspondence and its higher order analogues.

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Asymptotic Behavior of Wronskian Polynomials that are Factorized via p-cores and p-quotients

Bonneux

2020

Math Phys Anal Geom

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In this paper we consider Wronskian polynomials labeled by partitions that can be factorized via the combinatorial concepts of p-cores and p-quotients. We obtain the asymptotic behavior for these polynomials when the p-quotient is fixed while the size of the p-core grows to infinity. For this purpose, we associate the p-core with its characteristic vector and let all entries of this vector simultaneously tend to infinity. This result generalizes the Wronskian Hermite setting which is recovered when p = 2.

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Wronskian Appell polynomials and symmetric functions

Cited by 10 publications

References 32 publications

Coefficients of Wronskian Hermite polynomials

Coefficients of Wronskian Hermite polynomials

Quasi–invariant Hermite Polynomials and Lassalle–Nekrasov Correspondence

Asymptotic Behavior of Wronskian Polynomials that are Factorized via p-cores and p-quotients

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