Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics

We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we give the definition, the orthogonality relation, the three term recurrence relation and generating functions of all classes of orthogonal polynomials in this scheme. In chapeter 2 we give all limit relation between different classes of orthogonal polynomials listed in the Askey-scheme. 
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their definition, orthogonality relation, three term recurrence relation and generating functions. In chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials. Finally in chapter 5 we point out how the `classical` hypergeometric orthogonal polynomials of the Askey-scheme can be obtained from their q-analogues.

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The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, the exponential function, the probability integral and so on. Elliptic integrals proved to be the most important. They are connected with rectification of arcs of certain curves. The remarkable idea of Abel to replace these integrals by the corresponding inverse functions led to the creation of the theory of elliptic functions.

Representation of Lie groups and special functions

The two potentials for which a particle moving non-relativistically in a spherical space under the action of conservative central force executes closed orbits are found. When the curvature is zero they reduce to the familiar Coulomb and isotropic oscillator potentials of Euclidean geometry. The corresponding vector (for the former) and symmetric tensor (for the latter) constants of motion are constructed. For each system in N dimensions the Poisson bracket algebra in classical mechanics, and the commutator algebra in quantum mechanics, of these constants of motion and the angular momentum components are constructed. It is proved that by an appropriate choice of independent constants of motion these Poisson bracket algebras may be transformed into Lie algebraic structures, those of the symmetry groups SO(N+1) and SU(N) respectively: the Hamiltonian of each system is expressed as a function of the Casimir operators of its symmetry group. The corresponding transformations of the quantum mechanical commutator algebras are performed only for N=2: the corresponding expressions for the Hamiltonian as functions of the Casimir operators yield the energy levels of the two systems.

http://iopscience.iop.org/article/10.1088/0305-4470/12/3/006/pdf

Dynamical symmetries in a spherical geometry. I

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via “coupling constant metamorphosis” (or equivalently, via Stackel multiplier transformations). We present a table of the results.

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Superintegrable systems in Darboux spaces

This paper is part of a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in real or complex conformally flat spaces. Here we consider classical superintegrable systems with nondegenerate potentials in three dimensions. We show that there exists a standard structure for such systems, based on the algebra of 3×3 symmetric matrices, and that the quadratic algebra always closes at order 6. We show that the spaces of truly second-, third-, fourth-, and sixth-order constants of the motion are of dimension 6, 4, 21, and 56, respectively, and we construct explicit bases for the fourth- and sixth order constants in terms of products of the second order constants.

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Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory

A Hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion. This property has been extensively studied in the case of two-dimensional spaces of constant (possibly zero) curvature when all the independent integrals are either quadratic or linear in the canonical momenta. In this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions. This is done by examining in detail one of the spaces of revolution found by G. Koenigs. We determine that there are essentially three distinct potentials which when added to the free Hamiltonian of this space have this type of superintegrability. Separation of variables for the associated Hamilton–Jacobi and Schrodinger equations is discussed. The classical and quantum quadratic algebras associated with each of these potentials are determined.

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Superintegrability in a two-dimensional space of nonconstant curvature

This paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems, both classical and quantum, in conformally flat spaces. Here we study the Stackel transform (or coupling constant metamorphosis) as an invertible mapping between classical superintegrable systems on different spaces. Through the use of this tool we derive and classify for the first time all two-dimensional (2D) superintegrable systems. The underlying spaces are exactly those derived by Koenigs in his remarkable paper giving all 2D manifolds (with zero potential) that admit at least three second order symmetries. Our derivation is very simple and quite distinct. We also show that every superintegrable system is the Stackel transform of a superintegrable system on a constant curvature space.

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Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for the corresponding systems defined in Euclidean space and on the two‐dimensional sphere. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup Cartesian and polar coordinate cases, and the details of the structure of the quadratic algebras. We also study the polynomial eigenfunctions in elliptic coordinates of the n‐dimensional isotropic quantum oscillator.

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Ernie G. Kalnins

Papers

Superintegrable systems in Darboux spaces

Second order superintegrable systems in conformally flat spaces. III. Three-dimensional classical structure theory

Superintegrability in a two-dimensional space of nonconstant curvature

Second order superintegrable systems in conformally flat spaces. II. The classical two-dimensional Stäckel transform

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions