Special Issue on Symmetry, Separation, Super-integrability and Special Functions (S⁴)

The Guest Editors for this special issue are

Ernie Kalnins (University of Waikato, Hamilton, New Zealand)
Niky Kamran (McGill University, Montreal, Canada)
Peter Olver (University of Minnesota, Minneapolis, USA)
Pavel Winternitz (Universite de Montreal, Canada)

On September 17–19, 2010, the University of Minnesota hosted a conference honoring Willard Miller, Jr., on the occasion of his retirement. Entitled “Symmetry, Separation, Super-integrability and Special Functions” (S⁴ for short), the talks covered Willard's wide-ranging research interests, and featured most of his close friends, collaborators, students, and colleagues. Willard's many fundamental contributions and influence in these areas have been felt far and wide. Remarkably, while continuing to be incredibly active throughout his researcher career, which continues into retirement, he also served in many administrative capacities at the University of Minnesota, including Department Head, Associate Director and Director of the I.M.A., Associate Dean, and acting Dean. During the meeting, a scholarship fund for students in a combined Bachelors/Masters' program at Minnesota was established in Willard's honor. See the conference web site for full details: http://math.umn.edu/conferences/s4/.

The organizers decided it would be worth assembling a special issue of SIGMA devoted to the conference's mathematical themes, and agreed to serve as editors. The 21 papers gathered in this volume represent contributions closely related to many of the themes of research to which Willard has made fundamental contributions over the years, including separation of variables, special functions, integrable systems, and their associated geometric structures, all of which reflect the wide spectrum and the remarkable breadth of Willard's scientific interests. Thus, classical special functions, q-special functions and harmonic analysis appear in the contributions of Tom Koornwinder, Luc Vinet and Alexei Zhedanov, Kurt Bernardo Wolf and Luis Edgar Vicent, Amalia Torre, Howard Cohl; symmetry and separation of variables are the main theme of the papers by Antoni Sym and Adam Szereszewski, and Alberto Carignano, Lorenzo Fatibene, Ray McLenaghan and Giovanni Rastelli, Philip Broadbridge and Peter Vassiliou; classical and quantum integrable and super-integrable models are the substance of the papers by Alexander Turbiner, Yannis Tanoudis and Costas Daskaloyannis, Angel Ballesteros, Alberto Enciso, Francisco J. Herranz, Orlando Ragnisco and Danilo Riglioni, Claudia Chanu, Luca Degiovanni and Giovanni Rastelli, Sarah Post, Christiane Quesne, Ernie G. Kalnins, Jonathan M. Kress and Willard Miller, Jr.; finally the papers by Michael G. Eastwood and A. Rod Gover, Charles P. Boyer, Andrei A. Malykh and Mikhail B. Sheftel, Stephen C. Anco, Sajid Ali and Thomas Wolf, Serge Preston, Yuri Bozhkov and Peter J. Olver, are related to the more geometric aspects of Willard's interests, notably the role of symmetries and geometric structures in differential equations. The high quality of all these papers reflects the high esteem in which Willard is held by his friends and colleagues throughout the world. We would like to thank all the authors for these very fine contributions.

Papers in this Issue:

On Darboux's Approach to R-Separability of Variables

Antoni Sym and Adam Szereszewski
SIGMA 7 (2011), 095, 21 pages   [ abs   pdf ]

Appell Transformation and Canonical Transforms

Amalia Torre
SIGMA 7 (2011), 072, 34 pages   [ abs   pdf ]

From Quantum A_N (Calogero) to H₄ (Rational) Model

Alexander V. Turbiner
SIGMA 7 (2011), 071, 20 pages   [ abs   pdf ]

Exact Solutions of Nonlinear Partial Differential Equations by the Method of Group Foliation Reduction

Stephen C. Anco, Sajid Ali and Thomas Wolf
SIGMA 7 (2011), 066, 10 pages   [ abs   pdf ]

Balance Systems and the Variational Bicomplex

Serge Preston
SIGMA 7 (2011), 063, 18 pages   [ abs   pdf ]

The BGG Complex on Projective Space

Michael G. Eastwood and A. Rod Gover
SIGMA 7 (2011), 060, 18 pages   [ abs   pdf ]

Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on S²×S³

Charles P. Boyer
SIGMA 7 (2011), 058, 22 pages   [ abs   pdf ]

Symmetry Operators and Separation of Variables for Dirac's Equation on Two-Dimensional Spin Manifolds

Alberto Carignano, Lorenzo Fatibene, Raymond G. McLenaghan and Giovanni Rastelli
SIGMA 7 (2011), 057, 13 pages   [ abs   pdf ]

Pohozhaev and Morawetz Identities in Elastostatics and Elastodynamics

Yuri Bozhkov and Peter J. Olver
SIGMA 7 (2011), 055,  9 pages   [ abs   pdf ]

Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential

Yannis Tanoudis and Costas Daskaloyannis
SIGMA 7 (2011), 054, 11 pages   [ abs   pdf ]

The Fourier U(2) Group and Separation of Discrete Variables

Kurt Bernardo Wolf and Luis Edgar Vicent
SIGMA 7 (2011), 053, 18 pages   [ abs   pdf ]

The Role of Symmetry and Separation in Surface Evolution and Curve Shortening

Philip Broadbridge and Peter Vassiliou
SIGMA 7 (2011), 052, 19 pages   [ abs   pdf ]

On Parameter Differentiation for Integral Representations of Associated Legendre Functions

Howard S. Cohl
SIGMA 7 (2011), 050, 16 pages   [ abs   pdf ]

Superintegrable Oscillator and Kepler Systems on Spaces of Nonconstant Curvature via the Stäckel Transform

Ángel Ballesteros, Alberto Enciso, Francisco J. Herranz, Orlando Ragnisco and Danilo Riglioni
SIGMA 7 (2011), 048, 15 pages   [ abs   pdf ]

Recursions of Symmetry Orbits and Reduction without Reduction

Andrei A. Malykh and Mikhail B. Sheftel
SIGMA 7 (2011), 043, 13 pages   [ abs   pdf ]

On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials

Tom H. Koornwinder
SIGMA 7 (2011), 040,  8 pages   [ abs   pdf ]

First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

Claudia Chanu, Luca Degiovanni and Giovanni Rastelli
SIGMA 7 (2011), 038, 12 pages   [ abs   pdf ]

Models of Quadratic Algebras Generated by Superintegrable Systems in 2D

Sarah Post
SIGMA 7 (2011), 036, 20 pages   [ abs   pdf ]

Revisiting the Symmetries of the Quantum Smorodinsky-Winternitz System in D Dimensions

Christiane Quesne
SIGMA 7 (2011), 035, 21 pages   [ abs   pdf ]

A Recurrence Relation Approach to Higher Order Quantum Superintegrability

Ernie G. Kalnins, Jonathan M. Kress and Willard Miller Jr.
SIGMA 7 (2011), 031, 24 pages   [ abs   pdf ]

A Bochner Theorem for Dunkl Polynomials

Luc Vinet and Alexei Zhedanov
SIGMA 7 (2011), 020,  9 pages   [ abs   pdf ]