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

Gómez‐Ullate, David

doi:10.1088/1751-8121/aace4b

Cited by 15 publications

(29 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

“…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%

“…We identify Wronskian Hermite polynomials with Wronskians involving Laguerre polynomials. [35][36][37][38] The obtained identity is naturally labeled by the partition and its core and quotient; see Proposition 5. This generalizes well-known identities that interpret Hermite polynomials in terms of Laguerre polynomials; see (55) below.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Coefficients of Wronskian Hermite polynomials

2020

View full text Add to dashboard Cite

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.

show abstract

“…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%

“…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%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Coefficients of Wronskian Hermite polynomials

2020

View full text Add to dashboard Cite

show abstract

“…The pseudo‐Wronskian determinants in this paper can be regarded as a generalization of Jacobi–Trudi formulas that involve not just the partition λ or its conjugate partition

λ^{'}

, but also mixed representations described by the Durfee symbols introduced by Andrews in . Analogous results for the Laguerre and Jacobi classes, including appropriately defined pseudo‐Wronskians, will be presented in .…”

Section: Introductionmentioning

confidence: 96%

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

2018

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Trends in Supersymmetric Quantum Mechanics

David²

2019

Integrability, Supersymmetry and Coherent States

View full text Add to dashboard Cite

Along the years, supersymmetric quantum mechanics (SUSY QM) has been used for studying solvable quantum potentials. It is the simplest method to build Hamiltonians with prescribed spectra in the spectral design. The key is to pair two Hamiltonians through a finite order differential operator. Some related subjects can be simply analyzed, as the algebras ruling both Hamiltonians and the associated coherent states. The technique has been applied also to periodic potentials, where the spectra consist of allowed and forbidden energy bands. In addition, a link with nonlinear second-order differential equations, and the possibility of generating some solutions, can be explored. Recent applications concern the study of Dirac electrons in graphene placed either in electric or magnetic fields, and the analysis of optical systems whose relevant equations are the same as those of SUSY QM. These issues will be reviewed briefly in this paper, trying to identify the most important subjects explored currently in the literature. KeywordsSupersymmetric quantum mechanics • Coherent states • Painlevé equations • Painlevé transcendents • Polynomial Heisenberg algebras • Factorization method • Exact solutions • Spectral design • Graphene Dedicated to my dear friend and colleague Véronique Hussin. D. J. Fernández C ( )

show abstract

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

Cited by 15 publications

References 61 publications

Coefficients of Wronskian Hermite polynomials

Coefficients of Wronskian Hermite polynomials

Durfee Rectangles and Pseudo‐Wronskian Equivalences for Hermite Polynomials

Trends in Supersymmetric Quantum Mechanics

Contact Info

Product

Resources

About