# S4 Conference Abstracts

#### Reflections on special functions, results and problems

*Richard Askey (University of Wisconsin)*

*Abstract:* After having worked on special functions for over 50 years, there are a few surprising results which should be mentioned and some problems which have either not been solved or solved the right way. One surprising result is that there is a common extension of a^x*a^y = a^(x+y) and a^x*b^x = (ab)^x, which is a result with many free parameters which is almost 100 years old. The q-version is the minimal extension of the Rogers-Ramanujan identities with no quadratic powers in exponents. That is newer, only about 80 years old. Possible Lie and other algebraic methods might exist which would give a conceptual proof of the linearization formula for ultraspherical polynomials. Finally, there are some very attractive results which arise when considering basic hypergeometric series when q is a root of unity. Is there an algebraic or geometric structure which contains these as quantum groups help explain some of the results known for basic hypergeometric orthogonal polynomials and other basic series?

#### On toric contact geometry

*Charles P. Boyer (University of New Mexico)*

*Abstract:* A toric contact structure is a special case of a completely integrable Hamiltonian system. In this talk I first give a review of what is known about toric contact structures on compact manifolds. I then turn to describing toric contact structures on $S^3$ bundles over $S^2$. Based on joint work with J. Pati, the equivalence problem for such contact structures is discussed, and examples are given which are inequivalent as toric contact structures, but equivalent as contact structures. This phenomenon is related to the existence of distinct conjugacy classes of maximal tori in the contactomorphism group. On the other hand one can use certain discrete invariants to distinguish inequivalent contact structures. The simplest invariant is the first Chern class of the contact bundle, but a more powerful invariant is the contact homology of Eliashberg, Giventhal, and Hofer. We compute this contact homology for certain toric contact structures. Then if time allows I will discuss the problem of the existence of extremal Sasakian metrics related to these contact structures, and in particular, that of Sasaki-Einstein metrics.

#### Special solutions of the third and fifth Painlev\'e equations and vortex solutions of the complex Sine-Gordon equations

*Peter A. Clarkson (University of Kent, UK)*

*Abstract:* The Painlev\'e equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, which have arisen in a variety of physical applications. Further they may be thought of as nonlinear special functions. The general solutions of the Painlev\'e equations are transcendental, however they possess special solutions in the form of rational solutions and solutions expressible in terms of the classical special functions.

In this talk I shall discuss classical solutions of the third and fifth Painlev\'e equations. Rational solutions of the third and fifth Painlev\'e equations are expressed as wronskians of associated Laguerre polynomials. Special function solutions of the third and fifth Painlev\'e equations are expressed as wronskians in terms of Bessel functions and Whittaker functions, respectively. Using these solutions of the third and fifth Painlev\'e equations, the associated multi-vortex solutions of the complex Sine-Gordon equations in the plane will be derived.

#### Representation theory of ternary quadratic algebras. The case of the quantum generalized three dimensional Verrier-Evans system: algebraic calculations of the energy eigenvalues.

*Costas Daskaloyannis (Aristotle University of Thessaloniki, Greece)*

*Abstract:* In the three dimensional flat space any classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller ( J. Math. Phys. 48, 113518 (2007) ) have proved that, in the case of non degenerate potentials, i.e potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The exi\-stence of this sixth integral imply that the integrals of motion form a ternary quadratic Poisson algebra with five generators. The superintegrability of the generalized Kepler Coulomb potential that was investigated by Verrier and Evans (J. Math. Phys. 49, 022902 (2008)) is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the non degenerate case of systems with quadratic integrals. In this talk, the ternary quadratic associative algebra corresponding to the quantum Verrier-Evans system will be introduced. The structure, the Casimir operators and the the finite dimensional representation of this algebra will be discussed.

#### Differential equations for orthogonal polynomials.

*Mourad Ismail (City University of Hong Kong)*

*Abstract:* We discuss applications of the differential equations of polynomials orthogonal with respect to absolutely continuous measures. We show connection to discrete Painleve equations. Q-analogues will also be discussed.

#### Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials

*Tom H. Koornwinder (University of Amsterdam)*

*Abstract:* Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. They can be equivalently written as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a $2\times 2$ matrix-valued operator. As a new result made possible by this approach I obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little $q$-Jacobi polynomials also becomes possible in this way. Furthermore, limits to various types of non-symmetric $q$-Bessel functions will be considered. Corresponding limit algebras of the Askey-Wilson double affine Hecke algebra will be given as well. Tentatively, perspectives for the higher rank case will be discussed. This is joint work with Fehti Bouzeffour (Bizerte, Tunisia).

#### Higher order superintegrability

*Jonathan Kress (University of New South Wales)*

*Abstract:* Second order superintegrable systems have been intensively studied because of their connection with separation of variables. Recently, a family of systems proposed by Tremblay, Turbiner and Winternitz, has been shown to be superintegrable with higher order symmetries. A proof of the superintegrability of these and other similar systems will be discussed.

#### Symmetry operators and separation of variables for the Dirac equation on curved space

*Ray McLenaghan (University of Waterloo)*

*Abstract:* The status of the theory of separation of variables for the Dirac equation on two- and four-dimensional spin manifolds is reviewed. Special attention is given to the two-dimensional case where the theory is better developed. The structures of the symmetry operators that play an important role in the theory are described.

#### On new exact solutions for the Dirac-Pauli equation.

*Anatoly Nikitin (Institute of Mathematics of National Academy of Sciences)*

*Abstract:* A new exactly solvable relativistic model based on the Dirac-Pauli equation is presented. The model describes the Dirac fermion interacting with the electric field generated by a charged infinite filament and the magnetic field generated by a straight line current. It can be integrated using Lie symmetries and hidden supersymmetry with a matrix superpotential. In non-relativistic approximation the considered model is reduced to the integrable Pron'ko-Stroganov model.

#### Special functions and orthogonal polynomials of compact simple Lie groups

*Jiri Patera (Université de Montreal)*

*Abstract:* Any compact simple Lie group G of rank n give rise to several infinite families of orthogonal polynomials of n variables. Continuous and discrete orthogonality of the polynomials will be presented.

#### Superintegrability on the 3D hyperbolic $H^2_2\sim SO(2,2)/SO(2,1)$ space

*George Pogosyan (JINR, Dubna Russia and Yerevan State University, Yerevan, Armenia)*

*Abstract:* The work devoted to the investigation of the superintegrable systems on 3D hyperbolic $H_{2,1}\sim SO(2,2)/SO(2,1)$ space introduced in the paper of M.A. del Olmo, M.A.Rodriguez and P.Winternitz. Special attention is given to the generalized Kepler-Coulomb and oscillator problems with a centrifugal terms. Both potentials are analyzed in detail for separable systems of coordinates on this space. We have also constructed the analog of Runge-Lenz vector which allows us to build an algebra of symmetry, which is non linear. Furthermore, the analog of well-known from Euclidean space the Kustaanheimo-Steifel regularizing transformations for a quantum Kepler-Coulomb system are discussed. The work done in collaboration with David Petrosyan.

#### The coupling constant metamorphosis and superintegrable systems

*Sarah Post (Université de Montreal)*

*Abstract:* In this talk, I will discuss the coupling constant metamorphosis, also known as the St\"ackel transform, and its uses in studying integrable and superintegrable systems. I will review its role in the classification of second-order superintegrable systems and its generalization to higher-order systems. As an application, I will focus the relation between potentials with a harmonic oscillator or Kepler-Coulomb term. In particular, I will discuss two new families of superintegrable deformations of these potentials: the first was discovered by F. Tremblay, A.V. Turbiner and P. Winternitz and the latter by myself with P. Winternitz.

#### Algorithmic analysis of overdetermined systems of PDE: Symmetries, separation and historical perspectives

*Greg Reid (University of Western Ontario)*

My PhD with Ernie Kalnins was my introduction to the Kalnins-Miller research mixing geometry, symmetry, separation of variables and overdetermined systems of PDE. Throughout my career I have focused on overdetermined systems of PDE, and related algorithms. In this talk I will give an overview of algorithmic methods for over-determined systems of PDE, interspersed with historical comments on Willard

Miller's profound influence. Some new work on numerical algebraic-geometric work will also be discussed.

#### Classification of solvable Lie algebras - new approaches and developments

*Libor Snobl (Czech Technical University in Prague)*

*Abstract:* The classification of solvable Lie algebras will be reviewed, mostly from the perspective of solvable extensions of a given nilradical. This approach to the classification problem goes back to G. M. Mubarakzyanov who successfully employed it in his classification of 6-dimensional solvable Lie algebras. During past two decades it was applied by P. Winternitz and his collaborators, including myself, to classifications of solvable algebras of arbitrary finite dimension with a given structure of their nilradical, e.g. Abelian or Heisenberg. This general procedure together with its recent refinements will be reviewed and some of the classification results shown. Also, some new general results, e.g. an upper bound on dimension of solvable Lie algebras with the given nilradical, will be introduced.

#### Askey-Wilson polynomials and the ASEP

*Dennis Stanton (University of Minnesota )*

*Abstract:* The asymmetrical exclusion process (ASEP) is a discrete time Markov chain whose steady state probabilities are related to moments of the Askey-Wilson polynomials. Explicit formulas for these moments are given, along with a combinatorial interpretation in terms of tableaux. This is joint work with Sylvie Corteel, Richard Stanley, and Lauren Williams.

#### Some explicit solutions of the cable equation

*Sergei K. Suslov (Arizona State University)*

*Abstract:* We discuss explicit solutions of the cable equation for a general model of dendritic tree with tapering. A simple graphical approach to steady state solutions is also discussed. Joint work with Marco Herrera-Valdez.

#### Lie algebras and the Schr\"odinger equation

*Alexander Turbiner (UNAM, Mexico)*

*Abstract:* It is shown that all known solvable Schr\"odinger equations in $R^n$ which are invariant with respect to classical Weyl groups have a hidden algebra $gl(n+1)$. In turn, in the case of exceptional Weyl and Coxeter groups the hidden algebra is a certain infinite-dimensional, finitely-generated algebra.

#### Automorphisms of the Heisenberg – Weyl algebra and d-orthogonal polynomials

*Luc Vinet (Université de Montréal)*

*Abstract:* We show that the d-orthogonal Charlier polynomials appear as matrix elements of nonunitary transformations corresponding to automorphisms of the Heisenberg – Weyl algebra. Basic properties (duality, recurrence relations and difference equations) are derived from representations of the Heisenberg – Weyl algebra.

#### Separation of discrete coordinates

*Kurt Bernardo Wolf (Instituto de Ciencias Físicas Universidad Nacional Autónoma de México Cuernavaca)*

*Abstract:* We are interested in discrete Hamiltonian systems, in particular the finite harmonic oscillator whose position observables are { -j, -j+1,..., j }, living in the representation j of so(3), with N = 2j + 1 integer-spaced values. In the two-dimensional case, the mother algebra is so(4). With this we can describe N x N pixellated square images if we follow the chain so(4) = so(3) + so(3) reduced by so(2) + so(2); and images pixellated along circular coordinates if we follow the Gel'fand-Tsetlin chain so(4) reduced by so(3) and so(2). In the finite cartesian screen we have the finite analogue of Hermite-Gauss modes i.e. Kravchuk functions numbered by their Kravchuk polynomials; these can be gyrated (optical term) by an so(2) to the finite analogues of the angular Laguerre-Gauss modes. In the finite polar screen, we have Hahn functions, numbered with Hahn polynomials (which are the Clebsch-Gordan coefficients!) and discrete Fourier phases; there we can rotate images easily with another so(2). Combining them, we can unitarily rotate cartesian-pixellated images, and map images on a finite square screen to a circular one. Are discrete elliptic-pixellated screens possible?