A New Class of Orthogonal Polynomials: The Bessel Polynomials

We develop the formalism of quantum mechanics on three dimensional fuzzy space and solve the Schrödinger equation for a free particle, finite and infinite fuzzy wells. We show that all results reduce to the appropriate commutative limits. A high energy cut-off is found for the free particle spectrum, which also results in the modification of the high energy dispersion relation. An ultraviolet/infra-red duality is manifest in the free particle spectrum. The finite well also has an upper bound on the possible energy eigenvalues. The phase shifts due to scattering around the finite fuzzy potential well have been calculated.

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“…One can write the solutions of the above equation in terms of Bessel polynomials [10] g k,l (r) = e ±ikr kr y l ± i kr .…”

Section: Commutative Free Particlementioning

confidence: 99%

Spectrum of the three-dimensional fuzzy well

Chandra

Groenewald

Kriel

et al. 2014

J. Phys. A: Math. Theor.

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“…In the case of the Hua-Pickrell measures these are the pseudo-Jacobi polynomials and in the case of the Pickrell measures these are the Jacobi. The analogous role in this paper is played by the Bessel [21] and also Laguerre polynomials.…”

Section: Informal Introduction and Historical Overviewmentioning

confidence: 87%

“…Proposition 1.2 is proven in Section 2 as Proposition 2.3. A key role in the proof is played by the Bessel orthogonal polynomials [21]. Remark 1.3.…”

Section: The Inverse Wishart Measures M (ν) On Infinite Positive-defimentioning

confidence: 99%

Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices

Assiotis¹

2019

SIGMA

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The ergodic unitarily invariant measures on the space of infinite Hermitian matrices have been classified by Pickrell and Olshanski-Vershik. The much-studied complex inverse Wishart measures form a projective family, thus giving rise to a unitarily invariant measure on infinite positive-definite matrices. In this paper we completely solve the corresponding problem of ergodic decomposition for this measure.

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“…Following Krall and Frink, the generalized Bessel polynomials (GBPs) of degree n are defined by

ℬ_{n} false(x; α, β false) = true \sum_{k = 0}^{n} ((), \begin{matrix} center \\ array n \\ array k \end{matrix}) ((), \begin{matrix} center \\ array n + k + α - 2 \\ array k \end{matrix}) \frac{k!}{β^{k}} x^{k},

where n is a nonnegative integer, provided α is a real positive parameter and β is nonzero. When α = β = 2, the GBPs reduce to the proper Bessel polynomials (PBPs):

ℬ_{n} false(x; 2, 2 false) = true \sum_{k = 0}^{n} \frac{false(n + k false)!}{false(n - k false)! k!} {((), \frac{x}{2})}^{k} .

The first systematic study of PBPs and GBPs and their properties such as recurrence relations, generating function, normalizing factors, and the analogue of the Rodrigues' formula was conducted 20 years after by Krall and Frink in connection with the general solution of the wave equation in spherical coordinates. Shortly after Thomson discovered independently the BPs in his study of electrical networks so that nowadays these polynomials can be found not only in advanced research articles but also in textbooks of network synthesis and design .…”

Section: Introductionmentioning

confidence: 99%

“…The properties below cover only a part of the existing results of the GBPs, and a wider overview is contained in Krall and Frink . It is well known that the GBPs of degree n satisfy the second‐order differential equation :

x^{2} y^{''} + false(α x + β false) y^{'} - n false(n + α - 1 false) y = 0, n = 0, 1, 2, ⋯ .

If, however, β is zero, then the solution of Equation is

y (x) = A x^{n} + B x^{- (n + α - 1)},

where A and B are arbitrary constants.…”

Section: Introductionmentioning

confidence: 99%

On the construction of generalized monogenic Bessel polynomials

Abdalla

Abul-Ez

Morais

2018

Math Methods in App Sciences

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MSC Classification: 30G35; 33C45In recent years, much attention has been paid to the role of classical special functions of a real or complex variable in mathematical physics, especially in boundary value problems (BVPs). In the present paper, we propose a higher-dimensional analogue of the generalized Bessel polynomials within Clifford analysis via a special set of monogenic polynomials. We give the definition and derive a number of important properties of the generalized monogenic Bessel polynomials (GMBPs), which are defined by a generating exponential function and are shown to satisfy an analogue of Rodrigues' formula. As a consequence, we establish an expansion of particular monogenic functions in terms of GMBPs and show that the underlying basic series is everywhere effective.We further prove a second-order homogeneous differential equation for these polynomials.

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A New Class of Orthogonal Polynomials: The Bessel Polynomials

Cited by 85 publications

References 1 publication

Spectrum of the three-dimensional fuzzy well

Spectrum of the three-dimensional fuzzy well

Ergodic Decomposition for Inverse Wishart Measures on Infinite Positive-Definite Matrices

On the construction of generalized monogenic Bessel polynomials

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