Four-parameter potential box with inverse square singular boundaries

Alhaidari, A. D.; Taiwo, T. J.

doi:10.1140/epjp/i2018-11943-x

Cited by 2 publications

(2 citation statements)

References 16 publications

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“…This is different from all well-known orthogonal polynomials in the mathematics literature. However, in the physics literature and since 2005 a polynomial class with this property has been encountered frequently while solving various quantum mechanical problems [13][14][15][16][17][18][19]. Unfortunately, the analytic properties of this class of orthogonal polynomials (such as the weight function, generating function, orthogonality, asymptotics, etc.)…”

Section: Solutions Of the St 2 R 2 (B13a)mentioning

confidence: 99%

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

Alhaidari

2018

Journal of Mathematical Physics

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We introduce two ordinary second-order linear differential equations of the Laguerre-and Jacobi-type. Solutions are written as infinite series of square integrable functions in terms of the Laguerre and Jacobi polynomials, respectively. The expansion coefficients of the series satisfy threeterm recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Most of these are well-known polynomials whereas few are not. We present physical applications of these differential equations in quantum mechanics.MSC: 34-xx, 81Qxx, 33C45, 33D45

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Section: Solutions Of the St 2 R 2 (B13a)mentioning

confidence: 99%

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

Alhaidari

2018

Journal of Mathematical Physics

Self Cite

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“…Since 2005, however, we found a new class of exactly solvable problems that are associated with orthogonal polynomials, which were overlooked in the mathematics and physics literature [5][6][7][8][9][10][11][12]. These polynomials are defined, up to now, by their three-term recursion relations and initial value 0 ( ) 1 P   .…”

Section: Introductionmentioning

confidence: 99%

Open Problem in Orthogonal Polynomials

Alhaidari

2019

Reports on Mathematical Physics

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Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. It consists of a four-parameter polynomial with continuous spectrum on the whole real line and two of its discrete versions; one with a finite spectrum and another with countably infinite spectrum. A second class of these new orthogonal polynomials appeared recently while solving a Heun-type equation. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the Askey scheme of hypergeometric orthogonal polynomials. Up to now, these polynomials are defined by their three-term recursion relations and initial values. However, their other properties like the weight functions, generating functions, orthogonality, Rodrigues-type formulas, etc. are yet to be derived analytically. Due to the prime significance of these polynomials in physics and mathematics, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g., in terms of hypergeometric functions).

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Four-parameter potential box with inverse square singular boundaries

Abstract: Using the Tridiagonal Representation Approach, we obtain solutions (energy spectrum and corresponding wavefunctions) for a new five-parameter potential box with inverse square singularity at the boundaries.

Cited by 2 publications

References 16 publications

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

Open Problem in Orthogonal Polynomials

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