Special Issue "Symmetry in Orthogonal Polynomials"
A special issue of Symmetry (ISSN 2073-8994).
Deadline for manuscript submissions: closed (16 August 2016) | Viewed by 20366
Interests: harmonic analysis; representation theory; special functions of several variables; applications to mathematical physics, especially exactly solvable systems of quantum mechanics
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The concept of symmetry has been fundamental and studied for millennia. The ancient geometers already knew the five regular solids. For a long time, symmetry was a part of the discipline of geometry, but in more recent times it has become very important in analysis, mathematical physics, and of course, group theory. Symmetry is a key tool in analyzing functions of several variables. For example, the harmonic homogeneous polynomials, which are invariant under the group of rotations fixing the North Pole on the unit sphere in are essentially the same as Gegenbauer polynomials of index N/2−1. By now, this idea has been vastly generalized, for example, to interpreting Jacobi polynomials of several variables (defined on a simplex and orthogonal with respect to a Dirichlet measure) as harmonic polynomials with certain subgroup invariance properties.
In mathematical physics there are the quantum-mechanical models of Calogero–Moser–Sutherland type: N identical particles with 1/r2 interaction and possibly an external potential, of which wavefunctions involve Jack polynomials. The symmetric group occurs naturally in any system of identical particles, where the properties are invariant under the interchange of two particles. Recent developments have extended this to “supermodels” and introduced super polynomials with bosonic (commuting) and fermionic (anticommuting) variables. These are involved in the open question of whether SUSY (supersymmetry) manifests in the real world. Another application of orthogonal polynomials is as wavefunctions of isotropic quantum harmonic oscillators. Lie and quadratic algebras are being used to provide more insight into the Askey tableau, a scheme for organizing the classical polynomials of hypergeometric type.
Symmetry appears in algebraic combinatorics, for example in association schemes and distance-regular graphs. These structures are analyzed with the help of orthogonal polynomials, which arise as eigenfunctions of an associated Laplacian operator.
Symmetry in orthogonal polynomials also appears when the domain is a symmetric shape and the weight function is invariant under a group generated by reflections: for example orthogonal polynomials on a regular hexagon find an application in wave-front analysis for hexagonal mirror segments in large astronomical telescopes. Another example is the analysis of trigonometric polynomials, which are periodic on a lattice (or tesselation of space by regular polytopes).
In this Special Issue we aim to present the newest developments in the interaction of symmetry and orthogonal polynomials, in areas such as quantum physics, combinatorics, and classical analysis problems dealing with convergence of polynomial expansions. In the situations discussed above, much has been discovered, nevertheless, more needs to be done, in more precise formulas, approximation theorems about expansions in orthogonal polynomials of several variables, dependence on the parameters of a weight function, vanishing properties of specific polynomials (such as Jack and Macdonald), and so on.
Prof. Charles F. Dunkl
Manuscript Submission Information
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- weight functions invariant under reflection groups
- Calogero-Moser-Sutherland models
- Jack polynomials
- orthogonal polynomials in several variables of classical type
- polynomials periodic on a lattice