Admissibility condition for exceptional Laguerre polynomials

Durán, Antonio J.; Pérez, Mario

doi:10.1016/j.jmaa.2014.11.035

Cited by 23 publications

(44 citation statements)

References 14 publications

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“…We now introduce Wronskians involving Laguerre polynomials following Refs. . For any two partitions μ and ν with degree vectors

n_{μ}

and

m_{ν}

, and for any parameter

α \in R

such that the values

n_{1}, n_{2}, \dots, n_{ℓ (μ)}, m_{1} - α, m_{2} - α, \dots, m_{ℓ (ν)} - α

are pairwise different, we define the polynomial

{\overset{L}{true}}_{μ, ν}^{(α)} : = \frac{1}{Δ (n_{μ}, m_{ν} - α)} x^{(ℓ (μ) + α) ℓ (ν)} Wr false[f_{1}, f_{2}, \dots, f_{ℓ false(μ false)}, g_{1}, g_{2}, \dots, g_{ℓ false(ν false)} false],

where

\begin{matrix} true \\ right f_{j} (x) & left = {\hat{L}}_{n_{j}}^{false(α false)} (x), & left j = 1, 2, \dots, ℓ false(μ false), \\ right g_{j} (x) & left = x^{- α} {\hat{L}}_{m_{j}}^{false(- α false)} (x), & left j = 1, 2, \end{matrix}

…”

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

“…The Wronskian involving Laguerre polynomials appears in the setting of exceptional Laguerre polynomials . In both papers, polynomials

Ω_{μ, ν}^{false(α false)}

of degree

| μ | + | ν |

were introduced.…”

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

“…Similarly, for the Wronskian involving Laguerre polynomials, see or , it is proven in Refs. and that there are no zeros on the positive real line if and only if the parameter α and partitions μ and ν satisfy an admissibility condition. Via and , we can link both polynomials and therefore both conditions should be comparable.…”

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

“…Hence, a combinatorial interpretation of the admissibility condition in Refs. and in terms of quotients seems reasonable, but is omitted here as it does not fit in the scope of this paper.…”

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

“…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%

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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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“…We now introduce Wronskians involving Laguerre polynomials following Refs. . For any two partitions μ and ν with degree vectors

n_{μ}

and

m_{ν}

, and for any parameter

α \in R

such that the values

n_{1}, n_{2}, \dots, n_{ℓ (μ)}, m_{1} - α, m_{2} - α, \dots, m_{ℓ (ν)} - α

are pairwise different, we define the polynomial

{\overset{L}{true}}_{μ, ν}^{(α)} : = \frac{1}{Δ (n_{μ}, m_{ν} - α)} x^{(ℓ (μ) + α) ℓ (ν)} Wr false[f_{1}, f_{2}, \dots, f_{ℓ false(μ false)}, g_{1}, g_{2}, \dots, g_{ℓ false(ν false)} false],

where

\begin{matrix} true \\ right f_{j} (x) & left = {\hat{L}}_{n_{j}}^{false(α false)} (x), & left j = 1, 2, \dots, ℓ false(μ false), \\ right g_{j} (x) & left = x^{- α} {\hat{L}}_{m_{j}}^{false(- α false)} (x), & left j = 1, 2, \end{matrix}

…”

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

“…The Wronskian involving Laguerre polynomials appears in the setting of exceptional Laguerre polynomials . In both papers, polynomials

Ω_{μ, ν}^{false(α false)}

of degree

| μ | + | ν |

were introduced.…”

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

Section: Connection With Laguerre Polynomialsmentioning

confidence: 99%

“…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%

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Coefficients of Wronskian Hermite polynomials

2020

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Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

Gómez‐Ullate

2020

Orthogonal Polynomials

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These are the lecture notes for a course on exceptional polynomials taught at the AIMS-Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications that took place in Douala (Cameroon) from October 5-12, 2018. They summarize the basic results and construction of exceptional poynomials, developed over the past ten years. In addition, some new results are presented on the construction of rational solutions to Painlevé equation PIV and its higher order generalizations that belong to the A2n -Painlevé hierarchy. The construction is based on dressing chains of Schrödinger operators with potentials that are rational extensions of the harmonic oscillator. Some of the material presented here (Sturm-Liouville operators, classical orthogonal polynomials, Darboux-Crum transformations, etc.) are classical and can be found in many textbooks, while some results (genus, interlacing and cyclic Maya diagrams) are new and presented for the first time in this set of lecture notes. (2000). Primary 33C45; Secondary 34M55. Mathematics Subject Classification

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Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials

Simanek

2022

Comput. Methods Funct. Theory

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Admissibility condition for exceptional Laguerre polynomials

Cited by 23 publications

References 14 publications

Coefficients of Wronskian Hermite polynomials

Coefficients of Wronskian Hermite polynomials

Exceptional Orthogonal Polynomials and Rational Solutions to Painlevé Equations

Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials

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