Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a <i>q</i>-quadratic lattice

Tcheutia, Daniel Duviol

doi:10.1080/10236198.2019.1627346

Cited by 12 publications

(7 citation statements)

References 13 publications

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“…These programs were designed for older versions of maple and do not handle all orthogonal polynomials in the Askey table. Recently Tcheutia [14] extended the Maple implementation of retode to cover classical orthogonal polynomials on quadratic and q-quadratic lattices and was able to identify the second family (1.2) as Wilson polynomials, confirming the analysis in Section 3.…”

Section: Introductionsupporting

confidence: 61%

Solution of an Open Problem about Two Families of Orthogonal Polynomials

Assche

2019

SIGMA

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Section: Introductionsupporting

confidence: 61%

Solution of an Open Problem about Two Families of Orthogonal Polynomials

Assche

2019

SIGMA

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“…Since the labeling of orthogonal polynomials in the (q-)Askey scheme is by the sequences x k , h k , g k or by the parameters occurring in their expansions is so clean, these data may be helpful for recognizing polynomials in these schemes from the coefficients in the three-term recurrence relation, assuming that it would be possible to obtain these data from these coefficients. See Tcheutia [14] for recent work on this recognition problem by different methods.…”

Section: Further Perspectivesmentioning

confidence: 99%

“…3c. big q-Laguerre (11), q −1 -Meixner ( 13), affine q-Krawtchouk (16), quantum q −1 -Krawtchouk (14)…”

Section: Introductionunclassified

Charting the $q$-Askey scheme

Koornwinder

2021

Preprint

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Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the q-Askey scheme together with their q-hypergeometric representations by three sequences x k , h k , g k of Laurent polynomials in q k , two of degree 1 and one of degree 2, satisfying certain constraints. This gives rise to a precise classification and parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also describe the four-manifold structure underlying the scheme.

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“…, where   Substituting the value of  in terms of the energy from Eq. (19a), the PPS formula (24) could be inverted to give the following energy spectrum formula…”

Section: 8mentioning

confidence: 99%

“…are not yet known. This is still an open problem in orthogonal polynomials [21,23,24]. Nonetheless, it is conjectured that ( , ) ( ; , )…”

Section: 12mentioning

confidence: 99%

Bound states and the potential parameter spectrum

Alhaidari

Bahlouli

2020

Journal of Mathematical Physics

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In this article, we answer the following question: If the wave equation possesses bound states but it is exactly solvable for only a single non-zero energy, can we find all bound state solutions (energy spectrum and associated wavefunctions)? To answer this question, we use the "tridiagonal representation approach" to solve the wave equation at the given energy by expanding the wavefunction in a series of energy-dependent square integrable basis functions in configuration space. The expansion coefficients satisfy a three-term recursion relation, which is solved in terms of orthogonal polynomials. Depending on the selected energy we show that one of the potential parameters must assume a value from within a discrete set called the "potential parameter spectrum" (PPS). This discrete set is obtained from the spectrum of the above polynomials and can be either a finite or an infinite discrete set. Inverting the relation between the energy and the PPS gives the bound state energy spectrum. Therefore, the answer to the above question is affirmative.

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Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice

Cited by 12 publications

References 13 publications

Solution of an Open Problem about Two Families of Orthogonal Polynomials

Solution of an Open Problem about Two Families of Orthogonal Polynomials

Charting the $q$-Askey scheme

Bound states and the potential parameter spectrum

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