
MASCOS Workshop on Stochastics and Special Functions
The University of Queensland
Friday 22nd May 2009
General
Special functions crop up in all branches of mathematics, and in almost
all areas of application. They include the Gamma and Beta functions, Bessel,
Elliptic and Hypergeometric functions, and (famously) the Riemann zeta
function. There are classes of special functions whose members
are orthogonal (the inner product of distinct members is zero).
Orthogonal functions include spherical harmonics and Walsh functions,
but arguably the most important are the systems of
orthogonal polynomials, which include Chebyshev, Hermite,
Jacobi, Laguerre and Legendre polynomials.
This workshop, sponsored by the ARC Centre of Excellence
for Mathematics and Statistics of Complex Systems (MASCOS), concentrated on the
theory and applications of special functions with particular emphasis on
how they arise in stochastic processes. There were two main speakers:
Erik van Doorn, who spoke on birthdeath processes and extreme
zeros of orthogonal polynomials, and Peter Forrester, who spoke
on the connection between the zeros of the Riemann zeta function and
eigenvalues of random matrices. Additionally, there were five
shorter invited presentations.
Main speakers
 Erik van Doorn (University of Twente)
 Peter Forrester (University of Melbourne)
Invited speakers
 Richard Brak (University of Melbourne)
 Jan de Gier (University of Melbourne)
 Paul Leopardi (Australian National University)
 Peter Taylor (University of Melbourne)
 Ole Warnaar (University of Queensland)
Venue
The Riverview Room,
Emmanuel College,
St Lucia Campus, University of Queensland
[College map]
Organizers
Phil Pollett and Ole Warnaar (MASCOS and the University of Queensland)
Programme
Abstracts

Richard Brak
Combinatorial method for calculating perturbed Tchebycheff polynomials
Abstract:
I will give a lightning overview of the combinatorial formulation
of classical orthogonal polynomials. This understanding will
then be applied to the problem of computing perturbed Tchebycheff
polynomials. These polynomials satisfy the same threeterm recurrence
as Tchebycheff polynomials except for a finite set (the perturbation)
of the weights. These polynomials have applications in polymer phase
transitions and in computing the stationary state of the Asymmetric
Simple Exclusion Markov Process.

Erik van Doorn
On birthdeath processes and extreme zeros of orthogonal polynomials
Abstract:
The decay rate of a Markov process is an important quantity that
characterises the speed of convergence of the timedependent state
probabilities to their limiting values. In the specific setting of a
birthdeath process the decay rate can be identified with the smallest
point, or smallest point but one, in the support of the spectral
measure of the process. The latter is the orthogonalising measure for a
sequence of polynomials satisfying a threeterms recurrence relation with
coefficients that are determined by the parameters of the birthdeath
process.
During the last decade several new properties (bounds and positivity
criteria) of the decay rate of a birthdeath process have been obtained
by exploiting techniques that do not involve orthogonal polynomials. In
the talk I will show how these results can be translated to yield new
information on the smallest and largest zeros of orthogonal polynomials,
and on the support of the orthogonalising measure for an orthogonal
polynomial sequence, in terms of the coefficients in the threeterms
recurrence relation.

Peter Forrester
Zeros of the Riemann zeta
function, eigenvalues of random matrices and queueing
Abstract:
The statistical properties of prime numbers
have attracted the attention of many famous mathematicians.
Riemann introduced what is now referred to as the Riemann
zeta function for this purpose. That all the complex zeros of the
Riemann zeta function lie on a certain line is the celebrated
Riemann hypothesis. The computation of these Riemann zeros has
attracted a lot of attention, and a combination of
analytic and numerical calculations has revealed that the
large zeros have statistical properties which coincide with those
a large Hermitian random matrix. The latter are known in the theory
of chaotic quantum systems, giving weight to a spectral interpretation
of the Riemann zeros. Another surprising setting relating to the
eigenvalues of random Hermitian matrices is the distribution of
exit times for a fixed number of jobs begin processed by a large
number of servers. The purpose of this talk is to introduce these
topics, and to highlight what aspects of random matrix theory are
relevant to their study.

Jan De Gier
The asymmetric exclusion process and AskeyWilson polynomials
Abstract:
I will show how the steady state of the stochastic onedimensional
exclusion process with boundary reservoirs can be computed analytically
in terms of AskeyWilson polynomials. Special cases can be treated using
simpler degenerate polynomials such as the AlSalamChihara and
qHermite polynomials. I will further show how timedependent quantities
can be calculated using qPochhammer symbols.

Paul Leopardi
Polynomial interpolation on the unit sphere, reproducing kernels and
random matrices
Abstract:
This talk describes work in progress.
The setting of Sloan and Womersley for polynomial interpolation on the unit
sphere gives rise to a sequence of random Gram matrices. A random Gram matrix
for interpolation with degree of exactnesss t is determined by
(t+1)^{2}
independently uniformly distributed points on the sphere. Each entry of the
matrix is given by the evaluation of a kernel polynomial at the inner product
of a pair of these points. The kernel polynomials are scaled Jacobi
polynomials, which vary with the degree of exactness, but converge to a
function related to a Bessel function.
Relevant questions pertain to the distribution of eigenvalues and the
distribution of the determinant for a given finite degree, as well as
asymptotics of the eigenvalue distribution as the degree approaches infinity.
 Peter Taylor
The role of orthogonal polynomials in determining decay rates of
multidimensional queueing processes
Abstract:
In determining the set of possible decay rates of the stationary
distribution of a multidimensional queueing system we need to determine
the values of x for which a system of difference equations of the form
(1)
a(x) w^{n1} + b(x)
w^{n} + c(x)
w^{n+1},
with n ≥ 0, have positive solutions in
l_{1}.
Here a(x), b(x)
and c(x) are polynomials in x. Conditions for the solution of
equation (1) to be in
l_{1}
follow from elementary
considerations. In joint work with Dirk Kroese, Werner Scheinhardt and
Allan Motyer, I have used orthogonal polynomials to derive conditions
for positivity.
In this talk, I shall describe the context in which equation
(1) arises, and then go on to discuss how we ensure positivity
of the solutions.

Ole Warnaar
(q,t)Laguerre polynomials
Abstract: The (generalised) Laguerre polynomials are an important class of
polynomials, orthogonal on the positive halfline with respect to the
weight e^{x}
x^{a}.
They arise as solutions of Laguerre's differential
equation and play an important role in Gaussian quadrature. In this talk I
will describe a multivariable quantum extension of the Laguerre polynomials
by considering solutions of an ndimensional system of
qdifference equations.
Photos
(click for high resolution image)
Participants

Name 
Email 
Affiliation (domain) 





Vyacheslav Abramov 
vyacheslav.abramov at sci. 
Monash University (monash.edu.au) 

Ron Addie 
addie at 
University of Southern Queensland (usq.edu.au) 

Zdravko Botev 
botev at maths. 
University of Queensland (uq.edu.au) 

Richard Brak 
r.brak at ms. 
University of Melbourne (unimelb.edu.au) 

Tim Brereton 
tim.brereton at 
University of Queensland (uqconnect.edu.au) 

Darryn Bryant 
db at maths. 
University of Queensland (uq.edu.au) 

Fionnuala Buckley 
fbuckley at maths. 
MASCOS, University of Queensland (uq.edu.au) 

Robert Buttsworth 
bobb at maths. 
University of Queensland (uq.edu.au) 

Vivien Challis 
vchallis at maths. 
University of Queensland (uq.edu.au) 

Joshua Chan 
chancc at maths. 
University of Queensland (uq.edu.au) 

Robert Cope 
robert.cope at 
MASCOS, University of Queensland (uqconnect.edu.au) 

Erik van Doorn 
e.a.vandoorn at 
University of Twente (utwente.nl) 

Tony Downes 
downes at physics. 
University of Queensland (uq.edu.au) 

Murray Elder 
m.elder at maths. 
University of Queensland (uq.edu.au) 

Peter Forrester 
p.forrester at ms. 
University of Melbourne (unimelb.edu.au) 

Jan De Gier 
jdgier at 
University of Melbourne (unimelb.edu.au) 

Joseph Grotowski 
grotow at maths. 
University of Queensland (uq.edu.au) 

Nazer Halimi 
n_h_halimi at 
University of Queensland (yahoo.com.au) 

Sam Hambleton 
sah at maths. 
University of Queensland (uq.edu.au) 

Dirk Kroese 
kroese at maths. 
University of Queensland (uq.edu.au) 

Dejan Jovanovic 
dejan.jovanovic at 
MASCOS, University of Queensland (uqconnect.edu.au) 

Paul Leopardi 
paul.leopardi at 
Australian National University (anu.edu.au) 

Ross McVinish 
r.mcvinish at 
MASCOS, University of Queensland (uq.edu.au) 

Charles Meaney 
meaney at physics. 
University of Queensland (uq.edu.au) 

Gerard Milburn 
milburn at physics. 
University of Queensland (uq.edu.au) 

Eric Mortenson 
uqemorte at 
University of Queensland (uq.edu.au) 

Nur Idalisa Norddin 
nur.norddin at 
University of Queensland (uqconnect.edu.au) 

Daniel Pagendam 
pagendam at maths. 
MASCOS, University of Queensland (uq.edu.au) 

Phil Pollett 
pkp at maths. 
MASCOS, University of Queensland (uq.edu.au) 

Tony Roberts 
apr at maths. 
University of Queensland (uq.edu.au) 

Audrey Soedjito 
audrey.soedjito at 
University of Queensland (uqconnect.edu.au) 

Peter Taylor 
p.taylor at ms. 
MASCOS, University of Melbourne (unimelb.edu.au) 

Nimmy Thaliath 
n.thaliath at 
MASCOS, University of Queensland (uq.edu.au) 

Anand Tularam 
a.tularam at 
Griffith University (griffith.edu.au) 

Ole Warnaar 
o.warnaar at maths. 
University of Queensland (uq.edu.au) 

Bill Whiten 
W.Whiten at 
University of Queensland (uq.edu.au) 

YaoZhong Zhang 
yzz at maths. 
University of Queensland (uq.edu.au) 
