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Workshop Programme

for period 29 June - 3 July 2009

Discrete Systems and Special Functions

29 June - 3 July 2009

Timetable

Monday 29 June
08:30-09:55 Registration
09:55-10:00 Welcome - David Wallace
10:00-11:00 Ruijsenaars, S (Leeds/Loughborough)
  Hilbert-Schmidt operators vs. elliptic Calogero-Moser type systems Sem 1
 

This talk is concerned with elliptic Calogero-Moser quantum N-particle systems of nonrelativistic and relativistic type, associated with the Lie algebras A_{N-1} and BC_N. We outline various results from our ongoing programme to obtain suitable orthonormal joint eigenvector bases of the commuting Hamiltonians by employing special Hilbert-Schmidt operators. The integral kernels of the latter involve elliptic gamma functions in the relativistic (difference) case, and powers of theta functions in the nonrelativistic (differential) case.

 
11:00-11:30 Coffee
11:30-12:30 Ramani, A (École Polytechnique)
  Deautonomising integrable non-QRT mappings Sem 1
 

We examine a class of previously derived integrable mappings which do not belong to the QRT family and show that they can be extended to non-autonomous forms without losing the integrable character. We derive more non-QRT integrable mappings, obtain their non-autonomous froms and show how they can be integrated.

 
12:30-13:30 Lunch at Wolfson Court
14:00-15:00 Hereman, W (Colorado School of Mines)
  Symbolic Computation of Lax Pairs of Integrable Nonlinear Difference Equations on Quad-graphs Sem 1
 

The talk focuses on scalar 2-dimensional nonlinear partial difference equations (P-Delta-Es) which are completely integrable, i.e., they admit a Lax representation.

Based on work by Nijhoff, Bobenko and Suris, a method to compute Lax pairs will be presented. The method is largely algorithmic and can be implemented in the syntax of computer algebra systems, such as Mathematica and Maple.

A Mathematica program will be demonstrated that automatically computes Lax pairs for a variety of P-Delta-Es on quad-graphs, including lattice versions of the potential Korteweg-de Vries (KdV) equations, the modified KdV and sine-Gordon equations, as well as lattices derived by Adler, Bobenko, and Suris.

The symbolic computation of Lax pairs of nonlinear systems of integrable P-Delta-Es is work in progress. A few initial examples will be shown.

 
15:00-15:30 Tea
15:30-16:30 Hubert, E (INRIA Sophia Antipolis)
  Algebra and algorithms for the classification of differential invariants Sem 1
 

For a Lie group action known by its infinitesimals three practical descriptions of the differential algebra of differential invariants are given by generators and syzygies. The syzygies form a formally integrable system in terms of the non commuting invariant derivations. Applying a generalized differential elimination algorithm on those allows to reduce the number of generators. The normalized and edge invariants were the focus in the reinterpretation of the moving frame method by Fels & Olver (1999) and its application to symmetry reduction by Mansfield (2001). My contribution here is first to show the completeness of a set of syzygies for the normalized invariants that can be written down with minimal information on the group action (namely the infinitesimal generators). Second, I provide the adequate general concept of edge invariants and show their generating properties. The syzygies for edge invariants are obtained by applying the algorithms for differential elimination that I generalized to non-commuting derivations. Another contibution is to exhibit the generating and rewriting properties of Maurer-Cartan invariants. Those have desirable properties from the computational point of view. They are all the more meaningful when one understands that they are the differential invariants that come into play in the moving frame method as practiced by Griffiths (1974) and differential geometers. These contributions are easy to apply to any group action as they are implemented as the maple package AIDA, which works on top of the library DifferentialGeometry and the extension of the library diffalg to non commutative derivations.

 
16:30-17:30 Mansfield, E (Kent)
  On the moment map for smooth and discrete Hamiltonian systems Sem 1
 

The talk concerns Hamiltonian systems with a Lie group symmetry, that are conjugate to a variational problem. The motivation for the talk includes recent results of the speaker with Tania Goncalves, which show the structure of the set of conservation laws arising from Noether's Theorem in terms of invariants and a moving frame. We exhibit a simple explicit formula for the moment map for the related Hamiltonian system, in both smooth and discrete cases, and discuss the application of the Fels and Olver reformulation of the moving frame to the integration problem.

 
17:30-18:30 Wine reception
18:45-19:30 Dinner at Wolfson Court
Tuesday 30 June
09:00-10:00 Previato, E (Boston)
  Isomonodromy and integrability Sem 1
 

The property of having no movable critical points for an ordinary differential equation was linked with integrable systems via theta functions in the 19th century, and more recently, since the 1970s, with integrable partial differential equations via similarity reduction. A geometric integration of these features will be explored in the first part of the talk, after work by H. Flaschka (1980), which suggests a deformation of the spectral curve. This provides the segue to the second part of the talk, concerning a joint project with F.W. Nijhoff. The isomonodromy equations for spectral data (e.g., the Baker function) are studied as systems of ODEs, following R. Garnier (1912). Special functions, specifically the Kleinian sigma function, are implemented in the equations, to seek the Gauss-Manin-connection counterpart of the Legendre equation, by which R. Fuchs (1906) had connected the isomonodromy property and the absence of movable critical points. Work by Nijhoff et al. on discrete and Schwarzian equations would be related to this higher-genus Legendre version of the isomonodromy condition.

 
10:00-11:00 Mazzocco, M (Loughborough)
  Poisson brackets arising in Teichmuller theory and isomonodromic deformations Sem 1
 

In this talk we associate to the Teichmuller space of a disk with n marked points a Garnier system. We show that by clashing poles the Teichmuller space of an annulus with marked points is obtained and give constraints on the monodromy data in order to map the hole to a orbifold point. This construction leads to some interesting Poisson brackets on the geodesic length functions.

 
11:00-11:30 Coffee
11:30-12:00 Gordoa, PR (Universidad Rey Juan Carlos)
  Reduction of order for discrete systems Sem 1
 

By establishing a suitable Backlund correspondence, a reduction of order can be effected for discrete systems having a certain structure. In this way we succeed in extending to discrete systems ideas previously developed within the context of ordinary differential equations. As examples we give applications of our approach to discrete Painleve hierarchies.

 
12:00-12:30 Pickering, A (Universidad Rey Juan Carlos)
  Bäcklund transformations for a discrete second Painlevé hierarchy Sem 1
 

We present a new general formulation of our approach to obtaining auto-Backlund transformations. This approach is much more straightforward and can be applied directly to discrete equations. As an example, we give a derivation of auto-Backlund transformations for a discrete second Painleve hierarchy.

 
12:30-13:30 Lunch at Wolfson Court
14:00-14:30 Spicer, P (KL Leuven)
  A discrete system from the recurrence coefficients of 2-variable elliptic orthogonal polynomials Sem 1
 

A new class of orthogonal polynomials are considered from a formal approach: a family of two-variable orthogonal polynomials related through an elliptic curve. The formal approach means we are interested in those relations that can be derived, without specifying a weight function. Using generalized Sylvester identities, recurrence relations and bilinear relations between the recurrence coefficients are derived. These bilinear relations are shown to integrable in the sense that they have Lax pairs.

 
14:30-15:00 Dominici, D (State University of New York)
  Polynomial solutions of differential-difference equations Sem 1
 

We investigate the zeros of polynomial solutions to the differential-difference equation \[ P_{n+1}(x)=A_{n}(x)P_{n}^{\prime}(x)+B_{n}(x)P_{n}(x),~ n=0,1,\dots \] where $A_n$ and $B_n$ are polynomials of degree at most $2$ and $1$ respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlacing. Our result holds for general classes of polynomials but includes sequences of classical orthogonal polynomials as well as Euler-Frobenius, Bell and other polynomials.

 
15:00-15:30 Tea
15:30-16:30 Rahman, M (Carleton)
  On the probabilistic origin of some classical orthogonal polynomials in one and two variables Sem 1
 

We will give a brief review of the previous work by the author and his collaborators featuring a so-called " dynamic urn model" that gives rise to transition probability kernels whose eigenfunctions are some of the classical orthogonal polynomials, both discrete and continuous.A more recent investigation by essentially the same authors involves what we call "Cumulative Bernoulli Trials",leads to a kernel in two independent variables and four probability parameters. The eigenfunctions of this kernel have been shown to be the Krawtchouk polynomials in two variables. Connection of these polynomials with the 9-j symbols of Quantum Angular Momentum theory as well as their q-extensions will also be discussed.

 
16:30-17:30 Koelink, E (Radboud Universiteit Nijmegen)
  Huge formulas for simple special functions Sem 1
 

We recall some simple special functions of basic hypergeometric type, amongst others the q-Laguerre polynomials, q-Bessel functions and 2phi1-series. In the GNS-representation for the Haar weight for the quantum group analogue of the normaliser of SU(1,1) in SL(2,C) such functions occur as the matrix coefficients of the multiplicative unitary, and for this reason these functions occur as coefficients in certain expansions in the dual quantum group. The purpose of the talk is to show how such interpretations give rise to huge formulas for these functions, including big expansion formulas and some positivity results. These results are actually spin-off from the study of the dual quantum group, which is joint work with Wolter Groenevelt (Delft) and Johan Kustermans (Leuven).

 
18:45-19:30 Dinner at Wolfson Court
Wednesday 1 July
09:00-10:00 Rains, EM (CALTECH)
  Moduli spaces and elliptic difference equations Sem 1
 

Sakai's elliptic Painlevé equation is constructed via the geometry of certain rational algebraic surfaces. I'll discuss joint work with Arinkin and Borodin in which we interpret Sakai's surfaces as moduli spaces of second-order linear elliptic difference equations (and related objects), such that the elliptic Painlevé equation itself acts via isomonodromy transformations. The construction also lends itself to higher-order problems, and/or problems with more complicated singularity structure; I'll discuss what is known about the corresponding higher-dimensional moduli spaces.

 
10:00-11:00 Hone, A (Kent)
  Discretizations of Kahan-Hirota-Kimura type and integrable maps Sem 1
 

A few years ago, Hirota and Kimura found a new completely integrable discretization of the Euler top. The method of discretization that they used had already appeared in the numerical analysis literature, in the work of Kahan, who found an unconventional integration scheme for the Lotka-Volterra predator-prey system. Kahan's approach, as rediscovered by Hirota and Kimura, applies to any system of quadratic vector fields, and is consistent with a general methodology for nonstandard discretizations developed earlier by Mickens. Some new examples of integrable maps have recently been found using this method. Here we describe the results of applying this approach to integrable bi-Hamiltonian vector fields associated with pairs of compatible Lie-Poisson algebras in three dimensions, and mention some other examples (including maps from the QRT family, and discrete Painleve equations). This is joint work with Matteo Petrera and Kim Towler.

 
11:00-11:30 Coffee
11:30-12:00 Prellberg, T (QMUL)
  Area-perimeter generating functions of lattice walks: q-series and their asymptotics Sem 1
 

We present results on the generating function of a lattice model of an adsorbing polymer, realised as partially directed walks above a skewed line, weighted with respect to the number of contacts and the enclosed area. The solution is in form of a quotient of two q-Bessel functions, and the singular limit as q approaches 1 is of fundamental importance in the phase diagram of the model.

 
12:00-12:30 Tierz, M (Brandeis)
  Stieltjes-Wigert and quantum topological invariants Sem 1
 

We introduce the Stieltjes-Wigert polynomials and show their utility in the analytical computation of quantum topological invariants. Examples are given, the simplest being the computation of the Witten-Reshetikhin-Turaev invariant. The computation of quantum dimensions, presented in detail, requires an interesting mixture of Stieltjes-Wigert polynomials and key results borrowed from algebraic combinatorics. The relationship with random matrices and the relevance of other set of polynomials, such as the biorthogonal version of the Sieltjes-Wigert polynomials is also discussed.

 
12:30-13:00 Lunch at Wolfson Court
18:45-19:00 Dinner at Wolfson Court
Thursday 2 July
09:00-10:00 Stokman, J (Universiteit van Amsterdam)
  Bispectral quantum KZ equations and their solutions Sem 1
 

I will report on correspondences between bispectral problems for (1) Ruijsenaars' q-difference equations, (2) Cherednik's q-difference reflection equations, and (3) quantum Knizhnik-Zamolodchikov equations. In addition I will discuss the solutions to these bispectral problems, and their interrelations. An important role in this analysis will be played by Cherednik's q-analog of the spherical function, which solves the bispectral problem of Ruijsenaars' q-difference equations. This talk is partially based on joint work with Michel van Meer.

 
10:00-11:00 Schlosser, M (Universität Wien)
  Special commutation relations and combinatorial identities Sem 1
 

We study commutation relations involving special weight functions, for which we obtain a weight-dependent generalization of the binomial theorem. When the weight functions are suitably chosen elliptic (i.e., doubly-periodic meromorphic) functions, we have the notable special case of so-called "elliptic-commuting" variables (that generalize the q-commuting variables yx=qxy) satisfying an elliptic generalization of the binomial theorem. The latter is utilized to quickly recover Frenkel and Turaev's 10V9 summation formula, an identity fundamental to the (young) theory of elliptic hypergeometric series. Furthermore, the combinatorial interpretation of our commutation relations in terms of weighted enumeration of lattice paths allows us to deduce other combinatorial identities as well.

 
11:00-11:30 Coffee
11:30-12:00 Hay, M (Kyushu)
  A completeness study on a class of Lax pairs Sem 1
 

Despite the existence of a Lax pair for a given equation often being used as the definition of its integrability, recently there have been few studies that seek to find or categorize nonlinear equations that using Lax pairs as their starting point. Of those studies that do begin with Lax pairs, all choose a priori a form of the Lax pair, i.e. the type of dependence on the spectral parameter, thus limiting the possible results. We present a study that begins with Lax pairs for partial difference equations that are 2x2, where each entry of the Lax matrices contains only one separable term. Besides that the terms are general, no ansatz is made about the dependence on the spectral parameter, nor about the terms dependent on the lattice variables. We examine all possible groups of equations that can arise via the compatibility condition. Once a system of equations is extracted from the compatibility condition, it is solved in a way that preserves its generality, up to a point where nonlinear evolution equations are apparent, or the system is shown to be trivial. In fact, of all the potential Lax pairs identified by this method, only two lead to non-trivial and unconstrained evolution equations. These are new, higher order varieties of the lattice sine-Gordon (LSG) and the lattice modified KdV (LMKDV) equations. The remaining systems are shown to be trivial, overdetermined or underdetermined, which suggests a connection between the existence of a Lax pair and the singularity confinement method of identifying integrable equations. As we do not make any assumptions about the explicit dependence on the spectral parameter, we show that a particular nonlinear equation may have many Lax pairs, all depending on the spectral parameter in different ways. The effect that this freedom has on the process of inverse scattering is, as yet, unclear.

 
12:00-12:30 Pacharoni, I (Universidad Nacional de Córdoba)
  Matrix orthogonal polynomials and group representations Sem 1
 

In the scalar case, it is well known that classical orthogonal polynomials can be obtained as the zonal spherical functions of compact Riemannian symmetric space of rank one. The main purpose of this talk is to exhibit the interplay among the matrix orthogonal polynomials, the matrix hypergeometric functions and the matrix spherical functions in the context of the complex projective space. Also an example in two variables, related with the grassmannian manifolds will be considered.

 
12:30-13:30 Lunch at Wolfson Court
14:00-14:30 Atakishiyev, N (Universidad Nacional Autónoma de México)
  On integral and finite Fourier transforms of continuous q-Hermite polynomials Sem 1
 

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials of Rogers with respect to the classical Fourier integral transform. The behaviour of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed.

 
14:30-15:00 Yao, Y (Tsinghua)
  A new extended discrete KP hierarchy and its solutions Sem 1
 

We extend the discrete KP hierarchy (DKPH) by its symmetry restriction. This extended DKPH (exDKPH) includes the discrete KP equation with self-consistent sources (DKPESCS) as its first nontrivial equation. Lax representation of exDKPH is also pressented. The DKPESCS can be considered to be DKPE with non-homogeneous terms. By directly applying the method of variation of constant to the solutions of the DKPE obtained by Darboux transformation and its corresponding eigenfunctions, we construct the solutions of the DKPESCS.

 
15:00-15:30 Tea
15:30-16:30 Witte, NS (Melbourne)
  Deformation of the Askey-Wilson polynomials Sem 1
 

A $\mathbb{D}$-semi-classical weight is one which satisfies a particular linear, first order homogeneous equation in a divided difference operator $\mathbb{D}$. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first order homogeneous matrix equation in the divided difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case. The simplest examples of the $\mathbb{D}$-semi-classical orthogonal polynomial systems are precisely those in the Askey Table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the $\mathbb{D}$-semi-classical class it is entirely natural to define a generalisation of the Askey Table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from the deformations and their relations with the other elements of the theory. As an example we treat the first non-trivial deformation of the system defined by the highest divided difference operator, the Askey-Wilson operator, that is to say the Askey-Wilson polynomials.

 
16:30-17:30 Osinga, H (Bristol)
  Tangency bifurcations of global Poincaré maps Sem 1
 

A common tool for analyzing the qualitative behaviour of a periodic orbit of a vector field in R^n is to consider the Poincaré return map to an (n-1)-dimensional section. Poincaré used this technique to show instabilities in the solar system and Birkhoff continued these ideas to find a Poincaré map that gives information aobut the entire dynamics in the context of Hamiltonian systems. For general vector fields, particularly in experiments, people often choose an unbounded (n-1)-dimensional section of R^n and assume that the Poincaré map gives all the information about the dynamics. However, for such choices there will typically be points where the flow is tangent to the section. Such tangencies cause bifurcations of the Poincaré return map if the section is moved, even when there are no bifurcations in the underlying vector field. This talk discusses the interactions of invariant manifolds with the tangency loci on the section. Using tools from singularity theory and flowbox theory, we present normal forms of codimension-one tangency bifurcations in the neighbourhood of a tangency point. The study of these bifurcations is motivated by and illustrated with examples arising in applications.

This is joint work with Clare Lee (University of Strathclyde), Bernd Krauskopf (University of Bristol) and Pieter Collins (CWI, Amsterdam).

Tangency bifurcations of global Poincaré maps (abstract or preprint)
Clare M. Lee, Pieter J. Collins, Bernd Krauskopf & Hinke M. Osinga
SIAM Journal on Applied Dynamical Systems 7(3): 712-754, 2008.

Codimension-one tangency bifurcations of global Poincaré maps of four-dimensional vector fields (abstract or preprint)
Bernd Krauskopf, Clare M. Lee & Hinke M. Osinga
Nonlinearity 22(5): 1091-1121, 2009.

Dr Hinke Osinga Home Page: www.enm.bris.ac.uk/staff/hinke

 
19:00-23:00 Garden party at the Moller Centre
Friday 3 July
09:00-10:00 Its, A (Indiana University Indianapolis)
  Discrete Painlevé equations and orthogonal polynomials. Sem 1
 

Random matrices and orthogonal polynomials have been, for more than a decade, one of the principal sources of the important analytical ideas and exciting problems in the theory of discrete Painleve equations. In the orthogonal polynomial setting, the discrete Painleve equations appear in the form of the nonlinear difference relations satisfied by the relevant recurrence coefficients. The principal analytical question is the analysis of certain double-scaling limits of the solutions of the discrete Painleve equations. In the talk we will present a review on the subject using the Riemann-Hilbert formalism as a main analytic tool. The Riemann-Hilbert approach in the theory of discrete Painleve equations will be outlined as well.

 
10:00-11:00 van Assche, W (Katholieke Universiteit Leuven)
  Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials Sem 1
 

All classical orthogonal polynomials in the Askey table (and its q-extension) are orthogonal with respect to a weight w that satisfies a first order differential, (divided) difference or q-difference equation with polynomial coefficients of degree one and less than or equal to two respectively (Pearson equation). Their recurrence coefficients coefficients can be found using this Pearson equation by solving a first order difference equation. Semi-classical weights satisfy a Pearson equation with polynomial coefficients of higher degree. The recurrence coefficients then satisfy higher order difference equations. Many examples have been worked out and we will present some of the semi-classical weights that give rise to discrete Painlevé equations for the recurrence coefficients of the orthogonal polynomials. This is joint work with my student Lies Boelen.

 
11:00-11:30 Coffee
11:30-12:00 Hussain, MD (Veer Kunwar Singh)
  Some theorem on Eulerian integrals of the multivariable H-function and their applications Sem 1
 

The main object of the present paper is to derive key formulas for the fractional integeration of the multivariable H-function. Each of the general Eulerian integeral formulas are shown to wield interesting new results for various families of generalized hypergeometric functions of several variables. some of these applications of the key formulas would provide potentially useful generalizations of known results in the theory of fractional calculus.

 
12:00-12:30 Rappoport, J (Russian Academy of Sciences)
  Chebyshev polynomial approximations for some hypergeometric systems Sem 1
 

The hypergeometric type differential equations of the second order with polynomial coefficients and their systems are considered. The realization of the Lanczos Tau method with minimal residue is proposed for the approximate solution of the second order differential equations with polynomial coefficients. The scheme of Tau method is extended for the systems of hypergeometric type differential equations. A Tau method computational scheme is applied to the approximate solution of a system of differential equations related to the differential equation of hypergeometric type. The case of the discrete systems may be considered also. Various vector perturbations are discussed. Our choice of the perturbation term is a shifted Chebyshev polynomial with a special form of selected transition and normalization. The minimality conditions for the perturbation term are found for one equation. They are sufficiently simple for the verification in a number of important cases. Several approaches for the approximation of kernels of Kontorovich-Lebedev integral transforms--modified Bessel functions of the second kind with pure imaginary order and with complex order are elaborated. The codes of the evaluation are constructed and tables of the modified Bessel functions are published. The advantages of discussed algorithms in accuracy and timing are shown. The effective applications for the solution of some mixed boundary value problems in wedge domains are given.

 
12:30-13:30 Lunch at Wolfson Court
14:00-14:30 Ohyama, Y (Osaka)
  Analytic solutions to the q-Painlevé equations around the origin Sem 1
 

We study special solutions to the q-Painleve equations, which are analytic around the origin or the infinity. As the same as continuous Painleve equations, we have a finite number of such solutions. The q-Painleve equations are epressed as a connection preserving deformation (Jimbo and Sakai in q-PVI; M. Murata in other cases). We can determine the connection data for analytic solutions. In the case of q-PVI, the connection data reduces to Heine's basic hypergeometric functions.

 
14:30-15:00 Tsuda, T (Kyushu)
  UC hierarchy and monodromy preserving deformations Sem 1
 

The UC hierarchy is an extension of the KP hierarchy. The aim of this talk to explain how the UC hierarchy is related to the monodromy preserving deformation of Fuchsian ODE with certain spectral types, such as Painleve VI and the Garnier systems, through similarity reductions.

 
15:00-15:30 Tea
15:30-16:30 Kajiwara, K (Kyushu)
  Hypergeometric solutions to the symmetric discrete Painlevé equations Sem 1
 

The discrete Painlev\'e equations are usually expressed in the form of system of first-order ordinary difference equations, but it is possible to reduce them to single second-order difference equations by imposing certain conditions on parameters. The former generic equations are sometimes called ``asymmetric'', and latter ``symmetric'', referring to the terminology of the QRT mapping. A typical example is a discrete Painlev\'e II equation (dP$_{\rm II}$) \begin{displaymath} x_{n+1}+x_{n-1} = \frac{(an+b)x_n+c}{1-x_n^2}, \end{displaymath} and the ``asymmetric'' discrete Painlev\'e II equation (adP$_{\rm II}$) \begin{displaymath} Y_{n+1} + Y_{n} = \frac{(2a n+b)X_n + c+d}{1-X_n^2},\quad X_{n+1} + X_{n} = \frac{(a(2n+1)+b)Y_{n+1} + c-d}{1-Y_{n+1}^2}. \end{displaymath} dP$_{\rm II}$ is derived by imposing the constraint $d=0$ on adP$_{\rm II}$ and putting $X_n=x_{2n}$, $Y_n=x_{2n-1}$, respectively. adP$_{\rm II}$ arises as the B\"acklund transformation of P$_{\rm V}$, and hence its hypergeometric solutions are expressed by the Hankel determinant whose entries are given by the confluent hypergeometric functions. However, the above specialization does not yield the hypergeometric solutions to dP$_{\rm II}$ which are given by \begin{displaymath} x_n=\frac{2}{z}~\frac{\tau_{N+1}^{n+1}\tau_N^n}{\tau_{N+1}^n\tau_N^{n+1}}-1,\quad \tau_N^n=\det\left(H_{n+2i+j-3}\right)_{i,j=1,\ldots,N}. \end{displaymath} Here $H_n$ is the parabolic cylinder function satisfying \begin{displaymath} H_{n+1}-zH_n + nH_{n-1}=0, \end{displaymath} and $a=\frac{8}{z^2}$, $b=\frac{4(1+2N)}{z^2}$ and $c=-\frac{4(1+2N)}{z^2}$ ($N\in\mathbb{Z}_{\geq 0}$). More precisely, (i) the asymmetric structure of shifts in the determinant, and (ii) the entry $H_n$, cannot be recovered by putting $d=0$ in the hypergeometric solutions to adP$_{\rm II}$. Such ``inconsistency'' among the hypergeometric solutions to symmetric and asymmetric discrete Painlev\'e equations has been observed already in the first half of 90's, but left unsolved for a long time. Moreover, the determinant with similar asymmetric shift cannot be seen for the solutions to other integrable systems. In this talk, we consider the $q$-Painlev\'e equation of type $\widetilde{W}(A_2+A_1)^{(1)}$ ($q$-P$_{\rm III}$) as an example, and clarify the mechanism of the above phenomena by using the birational representation of the Weyl group. This work has been done in collaboration with N. Nakazono and T. Tsuda (Kyushu Univ.).

 
18:45-19:30 Dinner at Wolfson Court

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