|
Organizers |
The first addition formula and some of what came later
by
Richard Askey
Univ. of Wisconsin
The addition formulas for sine and cosine come from Ptolemy's theorem. Several proofs of Ptolemy's theorem will be given, including Euler's refinement of it and a 19th century extension will be given.
Date received: October 18, 2007
Multiple Hermite polynomials and some applications
by
Walter Van Assche
Katholieke Universiteit Leuven, Belgium
Hermite polynomials are well known orthogonal polynomials with respect to the
Gaussian weight w(x)=e-x2 on the real line. We consider multiple Hermite polynomials
H[n\vec] which are polynomials of degree |[n\vec]| = n1+n2+...+nr, for which
|
These multiple Hermite polynomials appear in a natural way when analysing random matrices with an external source and also in the analysis of non-interesecting Brownian motions. Both applications will be explained.
Date received: October 30, 2007
High-Precision Values of the Gamma Function of real argument
by
Ross Barnett
University of Waikato
Coauthors: J.A.Youngman
A method is described to calculate values of G(n), 0 ≤ n ≤ 1 to arbitrary precision by combining a Bessel function with a 0F1 function. Steed's algorithm is used to compute the regular Bessel function Jn(x), for a suitable argument x and real n, to arbitrary accuracy. Hence the gamma function is obtained. Example values are given to 200D. Verification is by the 80D-results of Fransén and Wrigge, by the use of the duplication formula, and by computing the closed-form results of Borwein and Zucker.
A caveat is offered concerning the coding of the Bessel functions in Numerical Recipes and in the GSL library.
Date received: October 27, 2007
Modified Bessel functions in Ramanujan's lost notebook
by
Bruce C. Berndt
University of Illinois at Urbana-Champaign
In his lost notebook, Ramanujan records several entries involving modified Bessel functions, although he does not use the standard definitions or notation for them. First, he states Koshliakov's formula, first published by N. S. Koshliakov in 1929. Second, he records Guinand's formula, first published by A. P. Guinand in 1955. Third, he offers a formula established by K. Soni in 1966. Fourth, he states three new formulas involving modified Bessell functions. However, most of the presentation will be devoted to a formula involving a double series of Bessel functions that the author cannot prove, but it is unclear if Ramanujan's claim is correct. If correct, the result is intimately related to the famous Dirichlet divisor problem.
Date received: October 24, 2007
The vanishing of the integral of the Hurwitz zeta function
by
Kevin A Broughan
Department of Mathematics, University of Waikato
The Hurwitz zeta function z(s, a) unifies the Riemann zeta function, used in the proof of the prime number theorem, and Dirichlet L-functions, used to show the infinitude of primes in arithmetic progressions, and can be used to derive their functional equations. In this talk I will show that the Riemann integral over the parameter range a ∈ (0, 1], whenever it exists, vanishes.
Date received: October 25, 2007
Congruences for Andrews-Paule's broken 2-diamond partition function.
by
Song Heng Chan
Nayang Technological University
In the latest of a series of papers on combinatorial investigations using a computer algebra package Omega, G. E. Andrews and P. Paule studied plane partitions of "hexagonal shape" and introduced broken k-diamond partitions as generalizations of the plane partitions. In this talk, we first give a brief introduction of the plane partitions leading up to the broken k-diamond partitions.
Next we sketch proofs of two conjectures of Andrews and Paule on congruences of broken 2-diamond partitions.
Date received: October 25, 2007
On The Nevanlinna Order Of Lommel Functions And Subnormal Solutions Of Certain Complex Differential Equations
by
Edmund Y. M. Chiang
Hong Kong University of Science and Technology
Coauthors: Kit-Wing Yu
In an earlier joint work with M. Ismail [Canadian J. Math. 58 (2006), 257-287] we investigated a class of homogeneous ordinary differential equations in the complex plane with Morse potential that can admit entire solutions with "small" Nevanlinna order of zeros in C if and only if it can be solved in terms of Bessel polynomials. We continue our study into a class of non-homogeneous ordinary differential equations in the complex plane and show it can a dmit "sub-normal solution" if and only if the solution can be written in terms of a composition of degenerated forms of Lommel or Struve functions and exponential function. New indentities and properties of the Lommel and the Struve functions are established.
Date received: November 11, 2007
Abel's Lemma on Summation by Parts and Theta Hyeprgeometric Series
by
Wenchang Chu
Department of Mathematics, Lecce University
Coauthors: Cangzhi Jia
The modified Abel lemma on summation by parts is systematically employed to review most of the identities of theta hypergeometric series.
Date received: October 31, 2007
Fourier Expansions of the Fundamental Solution for Powers of the Laplacian in Rn
by
Howard S. Cohl
Department of Mathematics, University of Auckland, New Zealand
Coauthors: Tom ter Elst
In this talk I will show how one can compute closed form algebraic expressions for the fundamental solution of the polyharmonic equation, i.e. for powers of the Laplacian, in Rn. These algebraic expressions can be used to compute Fourier expansions for the fundamental solutions of these operators by using well-known expansion formulae. I will show how the fundamental solutions for the polyharmonic equation naturally breaks up into two different classes in a finite set of separable hyper-spherical and hyper-cylindrical coordinate systems, i.e. those of even and odd dimensions. In odd dimensions I show how the coefficients for the expansions are given in terms of associated Legendre functions (toroidal harmonics) and in even dimensions I show how the coefficients can be given in terms of an interesting set of polynomials.
Date received: October 17, 2007
Asymptotic analysis of the Bell polynomials by the ray method
by
Diego Dominici
SUNY, New Paltz
We analyze the Bell polynomials Bn(x) asymptotically as as n→∞. We obtain asymptotic approximations from the differential-difference equation which they satisfy, using a discrete version of the ray method. We give numerical examples showing the accuracy of our formulas.
Date received: November 20, 2007
Does diffusion determine the drum?
by
Tom ter Elst
University of Auckland
Coauthors: W. Arendt (Ulm),
M. Biegert (Ulm)
The question of Kac is whether one can hear the shape of a drum. Or more precisely, whether all eigen frequencies of a drum determine the drum. In general the answer to the latter question is negative. The eigen frequencies are equal if and only if there exists a unitary operator which intertwines the corresponding Laplacians. In this talk we discuss what happens if the unitary operator is replaced by an order isomophism, i.e., if it maps positive functions to positive functions. Or equivalently, if the diffusion processes on the two drums are equal.
Date received: October 17, 2007
Some conjectures of Melham concerning representations by figurate numbers.
by
Michael Hirschhorn
UNSW
I will report on progress with proving a large number of conjectures recently published by Ray Melham concerning representations of a number as various combinations of figurate numbers.
Date received: October 30, 2007
Addition Theorems Via Continued Fractions
by
Mourad . H. Ismail
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
Coauthors: Jiang Zeng, Université de Lyon, Université Lyon 1,
Institute Camille Jordan, UMR 5028 du CNRS 69622 Villeurbanne
France
We show connections between a special type of addition formulas and a theorem of Stieltjes and Rogers. We use different techniques to derive the desirable addition formulas. We apply our approach to derive special addition theorems for Bessel functions and confluent hypergeometric functions. We also derive several additions theorems for basic hypergeometric functions. Applications to the evaluation of Hankel determinants are also given .
Date received: October 13, 2007
Orthogonal polynomials and associated algebras.
by
Ernie Kalnins
University of Waikato
Coauthors: Willard Miller,Jonathan Kress
In this talk we cover the the properties of the classical orthogonal polynomials. In particular we emphasize those properties that are in common with the polynomials obtained by separation of variables in Elliptic type coordinates.There is then some discussion of the use of representation theory to derive special function identities. Finally some intriguing use of classical mechanics is given which enables the properties of well known orthogonal polynomials to be put in context.
Date received: October 30, 2007
The zeros of the complementary error function
by
Andrea Laforgia
Dipartimento di matematica, Università di Roma3
Coauthors: Arpad Elbert
We show that the complementary error function erfc(z)has no zeros when Argz belongs to the interval [3p/4, p]
Date received: October 31, 2007
Sixteen Eisenstein Series
by
Heung Yeung Lam
Institute of Information and Mathematical Science, Massey University, Auckland, New Zealand
Coauthors: S.Cooper, Massey University
S. Ramanujan (1887 - 1920) gave fourteen families of series in his Second Notebook in Chapter 17, Entries 13 - 17. In each case he gave only the first few examples, giving us the motivation to find and prove a general formula for each family of series. In this talk, I will present a powerful tool (four versatile functions f0, f1, f2, and f3) to collect all of Ramanujan's example together.
Date received: October 25, 2007
Asymptotics for Gegenbauer-Sobolev and Hermite-Sobolev orthogonal polynomials associated with non-coherent pairs of measures
by
A. Sri Ranga
Universide Estadual Paulista, Campus de S.J. Rio Preto, Brazil
Coauthors: Cleonice F. Bracciali and Eliana X.L. de Andrade
Inner products of the type <p, q> = <p, q>s0 + <p’, q’>s1, where one of the corresponding measures s0 or s1 is the measure associated with the Gegenbauer (Hermite) polynomials, are usually referred to as Gegenbauer-Sobolev (Hermite-Sobolev) inner products. This presentation deals with some asymptotic relations associated with the orthogonal polynomials with respect to a class of Gegenbauer-Sobolev (Hermite- Sobolev) inner products. The inner products are such that the associated pairs of symmetric measures (s0, s1) are not within the concept of “symmetrically coherent pairs of measures”, introduced by Iserles et al in1991.
Date received: October 2, 2007
Macdonald polynomials in the light of basic hypergeometric series
by
Michael Schlosser
University of Vienna
We survey some (old and recent) results for Macdonald polynomials from a basic hypergeometric series point of view. This is helpful in the search for new identities involving Macdonald polynomials as they should correspond to known summation theorems for basic hypergeometric series.
Date received: October 29, 2007
Finite fields and (q, t)-binomials
by
Dennis Stanton
University of Minnesota
Coauthors: Vic Reiner
A (q, t)-binomial coefficient is defined, motivated by the invariant theory of the general linear group over a finite field. When either q (the finite field variable) or t (the Hilbert series variable) approaches 1, the result is the q-binomial coefficient. Several combinatorial interpretations, connections with Schur functions, and positivity results and conjectures will be discussed. Some inklings about generalized hypergeometric series will be proposed.
Date received: October 23, 2007
Permutable Polynomials and Rational Functions
by
Garry J. Tee
Department of Mathematics, University of Auckland
Many infinite sequences of permutable rational functions and some infinite sequences of permutable polynomials are constructed, on the basis of elliptic functions and trigonometric functions. Many identities connect those permutable rational functions.
Date received: October 23, 2007
Representations of certain binary quadratic forms as Lambert series
by
Pee Choon Toh
National University of Singapore
Classical algebraic number theory allows us to write certain Lambert series as q-series associated to classes of binary quadratic forms. We will recall how this is done and then give an ``inversion" process to represent some of these binary quadratic forms by Lambert series.
Date received: October 28, 2007
Tight frames of multivariate orthogonal polynomials
by
Shayne Waldron
Department of Mathematics, University of Auckland
Frame decompositions are useful because they are technically similar to orthogonal expansions (they simply have more terms) and can be constructed to have desirable properties that may be impossible for an orthogonal basis, e.g., in the case of wavelets certain smoothness and small support properties.
Here we show that frames are of interest for spaces of multivariate orthogonal polynomials where the desirable properties are symmetries of the weight (which an orthogonal basis cannot express). We present a number of (hopefully compelling) examples of such tight frames including multivariate Jacobi polynomials on a simplex and the orthogonal polynomials for a radially symmetric weight.
Date received: October 11, 2007
The Mukhin-Varchenko conjecture
by
Ole Warnaar
Department of Mathematics and Statistics, The University of Melbourne
In their work on the Knizhnik-Zamolodchikov equations, Mukhin and Varchenko were led to conjecture the existence of a Selberg integral for all simple Lie algebras. In this talk I will present a generalisation of the Selberg integral, thus proving the Mukhin-Varchenko conjecture for Lie algebras of type A.
Date received: November 11, 2007
Spread Polynomials
by
N J Wildberger
UNSW
Spread polynomials are a new family of orthogonal polynomials closely related to the Chebyshev polynomials, but with interesting number theoretical properties. They arise in universal geometry, a form of Euclidean geometry that holds over a general field, and the spread polynomials make sense in any field. We will describe remarkable factorization properties of these polynomials, connections with cyclotomy and applications to powers of rotations.
Date received: September 25, 2007
Semi-classical orthogonal polynomials and the Painlevé-Garnier systems
by
Nicholas Witte
University of Melbourne
Semi-classical deformations of the classical orthogonal polynomials are generically monodromy preserving systems of linear ODEs with respect to the deformation variables and define an important class of solutions to the Painlevé and Garnier equations. A scheme proposed by Sakai in 2001 organises the Painlevé equations, their discrete and q-difference analogs under a master elliptic Painlevé equation. It is possible to deform other orthogonal polynomials in the Askey scheme of hyper-geometric orthogonal polynomials in a semi-classical manner and derive the analogs of the Painlevé equations appearing in the Sakai scheme. The example of the Askey-Wilson system will be treated.
Date received: October 29, 2007