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Symmetry
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Symmetry in Special Functions and Orthogonal Polynomials
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Symmetry in Special Functions and Orthogonal Polynomials

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (1 June 2020) | Viewed by 17979

Special Issue Editors

Dr. Howard S. Cohl

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Guest Editor
Applied and Computational Mathematics Division, NIST, Mission Viejo, California, USA
Interests: special functions; orthogonal polynomials; q-series; fundamental solutions of elliptic PDEs on isotropic Riemannian manifolds
Prof. Dr. Charles F. Dunkl

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Guest Editor
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
Interests: harmonic analysis; representation theory; special functions of several variables; applications to mathematical physics, especially exactly solvable systems of quantum mechanics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Roberto S. Costas-Santos

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Guest Editor
Departamento de Métodos Cuandtitativos, Universidad Loyola Andalucia, Dos Hermanas, Sevilla, Spain
Interests: special functions; orthogonal polynomials; q-series; algebraic combinatorics; linear algebra and operators
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Hans Volkmer

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Guest Editor
Department of Mathematical Sciences, University of Milwaukee-Wisconsin, Milwaukee, Wisconsin, USA
Interests: ordinary and partial differential equations; spectral theory; special functions; mathematical statistics
Prof. Dr. Loyal Durand

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Guest Editor
Departmemt of Physics (Emeritus), University of Wisconsin-Madison, Madison, Wisconsin, USA
Interests: theoretical physics; mathematical physics; special functions; semigroups and addition formulas; fractional lie operators

Special Issue Information

Dear Colleagues,

Special functions, one of the oldest branches of real and complex analysis, have been exploited by Issac Newton, Gottfried Leibniz, Leonhard Euler, Carl Friedrich Gauss, Bernhard Riemann, and among many other great mathematicians, physicists, astronomers, scientists, and engineers. In the recent past, using many diverse methods, new special functions and orthogonal polynomials have been introduced and explored, new organizational structures have been discovered, and new results have been obtained for centuries-old special functions. In this Special Issue, we invite and welcome review, expository, and original research articles dealing with recent advances on the topics of special functions and orthogonal polynomials of one, as well as several, variables.

Dr. Howard S. Cohl
Prof. Dr. Charles F. Dunkl
Prof. Dr. Roberto S. Costas-Santos
Prof. Dr. Hans Volkmer
Prof. Dr. Loyal Durand
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Special functions
  • Orthogonal polynomials
  • q-series and q-calculus
  • Generalized, basic, elliptic, and Kaneko-Macdonald hypergeometric series
  • Addition theorems and eigenfunction expansions
  • Definite and indefinite integrals of special functions
  • Global analysis on Riemannian and pseudo-Riemannian manfiolds
  • Applications of special functions and orthogonal polynomials
  • Mathematical knowledge management of special functions and orthogonal polynomials

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (7 papers)

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22 pages, 345 KiB  
Article
Multi-Integral Representations for Associated Legendre and Ferrers Functions
by Howard S. Cohl and Roberto S. Costas-Santos
Symmetry 2020, 12(10), 1598; https://doi.org/10.3390/sym12101598 - 25 Sep 2020
Cited by 6 | Viewed by 2394
Abstract
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of [...] Read more.
For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero. Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
14 pages, 308 KiB  
Article
Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials
by Howard S. Cohl, Roberto S. Costas-Santos and Linus Ge
Symmetry 2020, 12(8), 1290; https://doi.org/10.3390/sym12081290 - 3 Aug 2020
Cited by 3 | Viewed by 3452 | Correction
Abstract
In this survey paper, we exhaustively explore the terminating basic hypergeometric representations of the Askey–Wilson polynomials and the corresponding terminating basic hypergeometric transformations that these polynomials satisfy. Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
11 pages, 335 KiB  
Article
Complex Asymptotics in λ for the Gegenbauer Functions C λ α ( z ) and D λ α ( z ) with z ∈ (−1, 1)
by Loyal Durand
Symmetry 2019, 11(12), 1465; https://doi.org/10.3390/sym11121465 - 1 Dec 2019
Cited by 5 | Viewed by 1742
Abstract
We derive asymptotic results for the Gegenbauer functions C λ α ( z ) and D λ α ( z ) of the first and second kind for complex z and the degree | λ | → ∞ , apply the results to [...] Read more.
We derive asymptotic results for the Gegenbauer functions C λ α ( z ) and D λ α ( z ) of the first and second kind for complex z and the degree | λ | → ∞ , apply the results to the case z ∈ ( − 1 , 1 ) , and establish the connection of these results to asymptotic Bessel-function approximations of the functions for z → ± 1 . Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
15 pages, 313 KiB  
Article
Addition Formula and Related Integral Equations for Heine–Stieltjes Polynomials
by Hans Volkmer
Symmetry 2019, 11(10), 1231; https://doi.org/10.3390/sym11101231 - 2 Oct 2019
Viewed by 2076
Abstract
It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special [...] Read more.
It is shown that symmetric products of Heine–Stieltjes quasi-polynomials satisfy an addition formula. The formula follows from the relationship between Heine–Stieltjes quasi-polynomials and spaces of generalized spherical harmonics, and from the known explicit form of the reproducing kernel of these spaces. In special cases, the addition formula is written out explicitly and verified. As an application, integral equations for Heine–Stieltjes quasi-polynomials are found. Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
21 pages, 6363 KiB  
Article
Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models
by Dana Černá
Symmetry 2019, 11(8), 999; https://doi.org/10.3390/sym11080999 - 3 Aug 2019
Cited by 5 | Viewed by 3402
Abstract
This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three [...] Read more.
This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i − j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model. Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
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14 pages, 321 KiB  
Article
Some Singular Vector-Valued Jack and Macdonald Polynomials
by Charles F. Dunkl
Symmetry 2019, 11(4), 503; https://doi.org/10.3390/sym11040503 - 7 Apr 2019
Viewed by 2055
Abstract
For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces [...] Read more.
For each partition τ of N, there are irreducible modules of the symmetric groups S N and of the corresponding Hecke algebra H N t whose bases consist of the reverse standard Young tableaux of shape τ . There are associated spaces of nonsymmetric Jack and Macdonald polynomials taking values in these modules. The Jack polynomials form a special case of the polynomials constructed by Griffeth for the infinite family G n , p , N of complex reflection groups. The Macdonald polynomials were constructed by Luque and the author. For each of the groups S N and the Hecke algebra H N t , there is a commutative set of Dunkl operators. The Jack and the Macdonald polynomials are parametrized by κ and q , t , respectively. For certain values of these parameters (called singular values), there are polynomials annihilated by each Dunkl operator; these are called singular polynomials. This paper analyzes the singular polynomials whose leading term is x 1 m ⊗ S , where S is an arbitrary reverse standard Young tableau of shape τ . The singular values depend on the properties of the edge of the Ferrers diagram of τ . Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)

Other

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2 pages, 190 KiB  
Correction
Correction: Cohl, H.S.; Costas-Santos, R.S.; Ge, L. Terminating Basic Hypergeometric Representations and Transformations for the Askey–Wilson Polynomials Symmetry 2020, 12, 1290
by Howard S. Cohl, Roberto S. Costas-Santos and Linus Ge
Symmetry 2020, 12(12), 2120; https://doi.org/10.3390/sym12122120 - 21 Dec 2020
Viewed by 1821
Abstract
The authors wish to make the following corrections to their paper [...] Full article
(This article belongs to the Special Issue Symmetry in Special Functions and Orthogonal Polynomials)
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