Workshop on Special Functions, FoCM'02, IMA, Minneapolis, 5-7 August 2002
This is one of the 19 workshops during the conference FoCM'02 at the IMA, University of Minnesota, Minneapolis, MN, USA, 5-14 August 2002. The workshop will run for 3 successive afternoons during 5-7 August (Monday - Wednesday).The workshop will be held immediately after the IMA 2002 Summer Program Special Functions in the Digital Age at the IMA in Minneapolis, 22 July - 2 August 2002.
Abstracts
Abstract of Berkovich' lecture
In this talk I explain how the Dyson adjoint of a partition can be
used in order to derive a binary tree of polynomial analogs of Euler's
pentagonal number theorem. This tree contains a well- known polynomial
identity due to Schur. At present only one of those new analogs has a
q-hypergeometric explanation. Next, I show how one can extend Dyson's
adjoint of partitions to deal with MacMahon's mod 2 graphs. This way one is
led to the new combinatorial proof of the Gauss identity. This talk is based
on my joint work with Frank Garvan.
(canceled)
Abstract of Bogatyrev's lecture
Many mathematical and physical theories (i.e. conformal field theory,
string theory, final gap solutions for completely integrable systems,
extremal polynomials) essentially use function theory on
Riemann surfaces. Such objects as abelian differentials, their integrals
and periods, quadratic differentials, Eichler periods, meromorphic
sections of vector bundles
over the curves etc. often appear in the formulas of theoreticians.
Sometimes it is necessary to know how the above entries vary with the
deformation of the curve, say if you have to solve some equations in
the moduli space of the curves. The latter problem arises in the
algebro-geometric approach to the Chebyshev optimization problems.
Every real algebraic curve admits Schottky uniformization with
convergent linear Poincaré theta series (A.I. Bobenko, 1986).
This theorem becomes a clue to the numerical analysis on Riemann
surfaces and their deformation spaces. We discuss effective calculation
of Schottky automorphic functions and effective solution of abelian
equations in Teichmüller space with application to the
extremal polynomials.
Abstract of Clarkson's lecture
The six Painlevé equations were first discovered around the turn
of the century by Painlevé and his colleague in an investigation of
nonlinear second-order ordinary differential equations. Recently there has
been considerable interest in the Painlevé equations primarily due to
the
fact that they arise as reductions of the soliton equations which
are solvable
by inverse scattering. Consequently the Painlevé equations can be
regarded
as completely integrable equations and possess solutions which can be
expressed in terms of solutions of linear integral equations, despite
being
nonlinear equations. Although first discovered from strictly mathematical
considerations, the Painlevé equations have arisen in a variety of
important physical applications including statistical mechanics, plasma
physics, nonlinear waves, quantum gravity, quantum field theory, general
relativity, nonlinear optics and fibre optics.
In this talk I shall discuss connection problems for the Painlevé
equations. Such problems can be characterized by the question as to how
the
asymptotic behaviours of solutions are related as the independent variable
is
allowed to pass towards infinity along different directions in the complex
plane. Connection problems have been previously tackled by a variety of
methods. Frequently these are based on the ideas of isomonodromic
deformation and the matching of WKB solutions, which is a nonlinear
version of the classical steepest descent method for oscillatory
Riemann-Hilbert problems and is rather complex. I shall describe a uniform
approximation method developed in collaboration with Bassom, Law &
McLeod.
This procedure, which is rigorous, removes the need to match solutions and
can
lead to simpler solutions of connection problems.
Abstract of Garrett's lecture
We will give a general theorem that arises from combinatorial consideration of
certain polynomials. We will state several corollaries. These corollaries
will be polynomial genralizations of many of the Rogers-Ramanujan type
identities from L.J. Slater's list.
Abstract of Howls' lecture
The Digital Library of Mathematical Functions is currently under
construction at NIST. One of the new types of special functions to be
included is that arising from canonical integrals with coalescing
saddles. Although studied previously as independent functions, such
integrals are the basis of applications of Thom's catastrophe theory to
a wide variety of problems, notably caustic diffraction phenomena in
wavefield (optics, quantum mechanics, waterwave, ...) and uniform
asymptotic analysis. These integrals fall into two types.
The first type are single dimensional integrals with (K+2)th order
polynomial phases (K saddlepoints) in the integrand, depending on real
parameters {x_{1}, x_{2}, ..., x_{K}}.
They form a natural hierarchy of special
functions with the most famous defined by K=2 being the Airy function
that decays exponentially at +infinity, and K=3 the Pearcey function. A
consequence of this hierarchy is that large-K functions can be expressed
asymptotically in terms of those with lower K-values.
The second type of canonical integrals studied are the elliptic and
hyperbolic umbilics, which take the form of double integrals and are
used to model more complicated diffraction phenomena.
In this talk we shall describe analytical properties of these classes of
functions and demonstrate methods to calculate such functions to high
numerical accuracies.
Abstract of Ismail's lecture
The Bethe Asatz equations are nonlinear algebraic equations
satisfied by the eigenvalues of a physical system. Stieltjes solved these
equations for the Coulomb gas model. This work is also connected to earlier
work of Heine who counted the number of polynomial solutions to second order
differential equations with polynomial coefficients. q-Analogues of these
results will be described and I will show the connection with Bethe Ansatz
equations for the XXX and XXZ models. In doing so one needs to develop a new
theory of singularities
of second order equations in the Askey-Wilson operators.
This is based on joint work with S. S. Lin and S.S. Roan from Taipei.
Abstract of Maier's lecture
The theory of quadratic and cubic hypergeometric transformations was
systematized by Goursat, who built on an earlier insight of Riemann.
A hypergeometric equation is a second-order Fuchsian equation on the Riemann
sphere with no more than three singular points, and a rational
transformation between two such equations is possible only if it takes
singular points to singular points, and appropriately transforms the
characteristic exponents of each singular point, i.e., the local monodromy.
The Heun equation (including its special case the Lamé equation) is a
Fuchsian equation with four rather than three singular points. The
coefficients of its series solutions, which may be called Heun series,
satisfy three-term rather than two-term recurrence relations. Also, these
solutions depend on an accessory parameter, besides the characteristic
exponents. So the Heun equation and its solutions are hard to handle. We
shall explain how, nonetheless, the Heun equation can be transformed to the
hypergeometric equation by a rational change of its independent variable,
so long as its parameters satisfy any of a family of equations. On the
level of series, this is equivalent to the existence of a family of
Heun-to-hypergeometric identities. The possible rational transformations
are reminiscent of the morphisms that occur in Klein's theory of pullbacks
of the hypergeometric equation, but Klein's theory is specific to
hypergeometric equations that have a full set of algebraic solutions, and
the new Heun-to-hypergeometric transformations are valid even in the
absence of algebraicity.
Abstract of Miller's lecture
We outline the basic ideas relating to the notion of superintegrable
potentials and how they are related to separability in multiple
coordinate systems. We give examples and indicate how superintegrability
can be of use, particularly in relation to resolving the degeneracy
problem for multiply degenerate bound states. Virtually all of the
classical special functions of mathematical physics (in one and several
variables) arise in this study, and formulas expanding one type of
special function as a series in another type emerge as a byproduct. We
describe how one can, in principle, classify all such systems, through
the use of symbolic algebra programs. This is joint work with E.G.
Kalnins, J. Kress and G. Pogosyan.
Abstract of Olde Daalhuis' lecture
In this talk we will present asymptotic expansions for the Gauss
hypergeometric function
F(a+e_{1} p,b+e_{2} p;c+e_{3} p;-z), as |p| goes to
infinity,
where e_{j} are elements of {-1,0,1}.
The expansions hold for fixed values of a, b, c, and are uniformly valid
for z in
the domain |ph z|<pi.
Abstract of Petkovsek's lecture
Wilf and Zeilberger conjectured in 1992 that a hypergeometric
term is proper-hypergeometric if and only if it is holonomic. Their
conjecture concerns hypergeometric terms which depend on several discrete
and continuous variables. Jointly with S.A. Abramov, we have proved a
slightly modified version of their conjecture in the multivariate discrete
case, namely that every holonomic hypergeometric term is conjugate to a
proper term (meaning that they have the same certificates). This
modification is necessary as shown, e.g., by the bivariate hypergeometric
term T(n,k) = |n-k| which is holonomic but not proper. Our proof is based
on the Ore-Sato Theorem which states essentially that for every
hypergeometric term T there is a rational function R and a proper term T'
such that T and RT' have the same certificates. This was proved in the
bivariate case by Ore using elementary means, and in the multivariate case
by Sato using homological algebra. We give an elementary proof of the
multivariate Ore-Sato Theorem. The necessary tools that are useful also for
other purposes are normal forms of rational functions and several notions
of shift-invariance for multivariate polynomials. We have also shown that a
rational sequence is holonomic if and only if its denominator factors into
integer-linear factors.
Abstract of Schlosser's lecture
We present a new matrix inverse with applications in the
theory of bilateral basic hypergeometric series. Our matrix
inversion result is directly extracted from an instance of
Bailey's very-well-poised _{6}psi_{6} summation theorem, and
involves two infinite matrices which are not lower-triangular.
We combine our bilateral matrix inverse with known basic
hypergeometric summation theorems to derive, via inverse
relations, several new identities for bilateral basic
hypergeometric series.
Abstract of Spiridonov's lecture
An elliptic extension of the WP-Bailey chain of Andrews
is constructed. This leads to an infinite sequence of
identities for very-well-poised elliptic hypergeometric
series. In particular, a new proof of Frenkel-Turaev's
elliptic analogue of the Bailey's identity is obtained.
Elliptic extention of a Bressoud's Bailey pair results
in a new elliptic hypergeometric series identity
representing a generalization of one of the identities
for very-well-poised basic hypergeometric series derived
recently by Andrews and Berkovich.
Abstract of Temme's lecture
A new method will be presented for obtaining convergent
expansions for Bernoulli, Charlier, Laguerre and Jacobi
polynomials (and perhaps other special functions). The expansions
have an asymptotic character for large values of the degree of
the polynomials. The expansions are obtained from Cauchy-type
integrals, that follow from the generating functions. For some of
the results we need two-point Taylor expansions for analytic
functions. A few properties of these expansions will be
discussed as well. This research is done together with José L.
Lopez (Pamplona, Spain).
Abstract of Terwilliger's lecture
Let K denote a field, and let
V denote a
vector space over K with finite positive dimension.
By a Leonard pair on V we mean
a pair of linear transformations A : V -> V
and B : V -> V
which satisfy
conditions (i), (ii) below:
(i) There exists a basis for V with respect to which
the matrix representing A is irreducible tridiagonal and
the matrix representing B is diagonal.
(ii) There exists a basis for V with respect to which
the matrix
representing B is irreducible tridiagonal and
the matrix representing A is diagonal.
(A tridiagonal matrix is said to be irreducible whenever
each entry on the superdiagonal or subdiagonal
is nonzero.)
The Leonard pairs
are closely related to the q-Racah
polynomials and related
polynomials in the Askey Scheme. Roughly speaking,
it is appropriate
to think of a Leonard pair as a sequence of q-Racah polynomials
in disguise.
In this talk we discuss several "canonical forms" for
Leonard pairs. We call these the
TD-D canonical form
and the LB-UB canonical form.
In the TD-D canonical form, the Leonard pair is represented by
an irreducible tridiagonal matrix and a diagonal matrix, subject
to a certain normalization. In the
LB-UB canonical form, the Leonard pair is represented by
a lower bidiagonal matrix and an upper bidiagonal matrix, subject
to a certain normalization. We show each Leonard pair is isomorphic
to a unique Leonard pair which is in TD-D canonical form and
a unique Leonard pair which is in LB-UB canonical form.
We give a detailed description of each of these forms.
References
[1]
T. Ito, K. Tanabe and P. Terwilliger,
Some algebra related to P- and Q-polynomial association
schemes,
in
Codes and Association Schemes (Piscataway NJ, 1999),
Amer. Math. Soc., 2000.
[2]
P. Terwilliger,
Two linear transformations each tridiagonal with respect to an
eigenbasis of the other,
Linear Algebra Appl. 330 (2001), 149-203.
[3]
P. Terwilliger,
Two relations that generalize the q-Serre relations and the
Dolan-Grady relations,
in
Proceedings of Nagoya 1999 Workshop on Physics and
Combinatorics (Nagoya, Japan 1999),
World Scientific, 2000.
[4]
P. Terwilliger,
Leonard pairs from 24 points of view,
accepted by Rocky Mountain J.
(special issuse, Special functions, Tempe, Arizona, 2000).