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- September 15, 1999 -
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- O P - S F N E T Volume 6, Number 5 -
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- Editor: -
- Martin Muldoon muldoon@yorku.ca -
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- The Electronic News Net of the SIAM Activity Group -
- on Orthogonal Polynomials and Special Functions -
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Today's Topics
1. From the Editor
2. NATO ASI: Special Functions 2000
3. Workshop on Quantum Groups, Morelia, Mexico
4. Symposium on Trends in Approximation Theory
5. International Symposium on Applied Mathematics,
6. OPSFA-Patras
7. Hong Kong Panel Discussion
8. Errors in Andrews-Askey-Roy
9. Proceedings "Algebraic Methods and q-Special Functions"
10. Proceedings "Applications and Computation of Orthogonal
Polynomials"
11. Proceedings "Continued Fractions: from analytic number theory to
constructive approximation"
12. Askey featured in SIAM News article
13. NSF Funding for NIST Digital Library Project
14. SIAM Student Paper Prizes
15. SIAM Student Travel Awards
16. Call for Nominations/Polya Prize
17. OP-SF preprints in xxx archive
18. Changes of Address, WWW Pages, etc
19. Subscribing to OP-SF NET
20. Obtaining back issues of OP-SF NET and submitting contributions
to OP-SF NET and Newsletter
Calendar of Events:
1999
September 14-18: International Conference on Analytic Methods of
Analysis and Differential Equations, Minsk, Belarus 5.6 #6
September 20-24: International Symposium on Orthogonal Polynomials,
Special Functions and Their Applications, Patras, Greece
5.4 #3 6.1 #8 6.5 #6
October 31 - November 7: Benin Workshop on Contemporary
Problems in Mathematical Physics 6.3 #6
November 8-12: Hong Kong Workshop on Minimal Energy Problems 6.3 #5
November 11-13: Conference on Symbolic Computation, Number Theory,
Special Functions, Physics and Combinatorics,
Gainesville, Florida, USA 6.4 #3
2000
January 5-7: Workshop on Computational Algebraic Analysis
Berkeley, California, USA 6.4 #5
January 10-14: Symposium on Asymptotics and Applied Analysis
San Diego, California, USA 6.4 #6
March 27-31: Workshop on Quantum Groups, Morelia, Mexico 6.5 #3
May 17-20: Symposium on Trends in Approximation Theory,
Nashville, Tennessee, USA 6.5 #4
May 29 - June 9: Special Functions 2000: Current Perspective
and Future Directions, Tempe, Arizona, USA 6.5 #2
July 3-7: Alhambra 2000, a Joint Mathematical European-Arabic
Conference 6.4 #7
July 10-14: SIAM Annual Meeting in Puerto Rico
July 17-22: I Colloquium on Lie Theory and Applications,
Vigo, Spain 6.4 #8
July 24-29: Summer School "Orthogonal Polynomials and Special
Functions", Laredo, Spain.
August 14-18: International Symposium on Applied Mathematics,
Dalian, China 6.5 #5
Topic #1 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: From the Editor
With about thirty MAA members, I enjoyed five days of Calculus, Counting
and cool weather in Duluth, Minnesota last month. The occasion was a
seminar organized by the North Central Section of the MAA and the main
speaker was Richard Askey, well-known to readers of this publication.
Askey presented 10 lectures on the topic "Calculus and Counting" with
titles like "Fermat and the q-calculus", "Permutations and Inversions",
"The Binomial Theorem", "The Normal Integral","Partitions", "Rational
functions and rearrangements of balls in boxes", "A problem of Stieltjes".
The purpose was instructional and although much of the material was
somewhat known to readers Askey's expository work (including the
Andrews/Askey/Roy book Special Functions; see Topic #8) the delivery was
in Dick's inimitable style, laced with anecdotes generated by his
well-known concern for educational matters. He even gave an eleventh
(public) lecture on Ramanujan. The promised cool weather did materialize
to the pleasure of participants. Even the torrential rain was welcomed by
those escaping the usual stifling heat of a North American summer. The
local arrangements were impeccably done and the participants enjoyed the
excellent new facilities of the Mathematics Department at the University
of Minnesota, Duluth, "a Great University by a Great Lake" as the
publicity aptly puts it.
In this issue we we publish (Topic #7) a report on the Panel Discussion
held at the Hong Kong Workshop in June. The approach of the new
millennium makes it natural to look backward and forward. A similar
discussion is scheduled for OPSFA in Patras this month. (See Topic #6)
And in May-June 2000, a NATO Advanced Study Institute with the title
"Special Functions 2000: Current Perspective and Future Directions" is
scheduled for Tempe, Arizona (See Topic #2 for preliminary information.)
To judge by Topic #17 in this issue, workers in our areas are increasingly
using the xxx archive to deposit copies of their current and earlier work.
Thanks to Roelof Koekoek for setting a good recent example! In this issue,
we experiment by including more information including abstracts and e-mail
addresses for the papers which we mention. I would welcome feedback on
the desirability of continuing to do this. One can foresee a time when
web access will be so universal that it will not be necessary to have this
alerting service.
Topic #2 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Sergei Suslov
Subject: NATO ASI: Special Functions 2000
This is preliminary information from the ASI web site:
http://math.la.asu.edu/~sf2000/
NATO Advanced Study Institute
Special Functions 2000:
Current Perspective and Future Directions
Arizona State University
Tempe, Arizona, U.S.A.
May 29 to June 9, 2000
Objective of the ASI: to summarize results in special functions and their
diverse applications obtained over the last 3 decades, and to discuss
future directions.
International Organizing Committee:
Sergei Suslov, Director from NATO country
Arizona State University, U.S.A.
Vyacheslav Spiridonov, Director from Partner country
Joint Institute for Nuclear Research, Dubna, Russia
Tom Koornwinder, KdV Institute, University of Amsterdam, The Netherlands
Luc Vinet, McGill University, Montreal, Canada
Local Organizing Committee:
Joaquin Bustoz, Chair, Arizona State University
Mourad Ismail, University of South Florida
Sergei Suslov, Arizona State University
Lecturers:
G. Andrews, Pennsylvania State University, U.S.A.
R. Askey, University of Wisconsin, Madison, U.S.A.
P. Deift, Courant Institute, U.S.A.
C. Dunkl, University of Virginia, U.S.A.
A. Grunbaum, University of California, Berkeley, U.S.A.
M.E.H. Ismail, University of South Florida, Tampa, U.S.A.
A. Its, Indiana University - Purdue University, Indianapolis, U.S.A.
E. Koelink, Technische Universiteit Delft, The Netherlands
T. Koornwinder, KdV Institute, University of Amsterdam, The Netherlands
I. Macdonald, Queen Mary College, London, England (not confirmed)
S. Milne, The Ohio State University, U.S.A.
O. Njastad, Norwegian University of Science & Technology, Norway
M. Rahman, Carleton University, Ottawa, Canada
V. Spiridonov, Joint Institute of Nuclear Research, Dubna, Russia
D. Stanton, University of Minnesota, U.S.A.
S. K. Suslov, Arizona State University, U.S.A.
N. Temme, CWI, Amsterdam, The Netherlands
V. N. Tolstoi, Moscow State University, Russia
L. Vinet, McGill University, Montreal, Canada
A. Zhedanov, Donetsk Institute for Physics and Technology, Ukraine
Address:
Advanced Study Institute SF2000
Arizona State University
Department of Mathematics
Box 871804
Tempe, AZ 85287-1804
U.S.A.
E-mail: sf2000@math.la.asu.edu
Fax: 1-480-965-8119
Web page: http://math.la.asu.edu/~sf2000/
Topic #3 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: Workshop on Quantum Groups, Morelia, Mexico
[This information is taken from the web site:
http://msri.org/activities/events/9900/qgroups/]
Workshop on Quantum Groups
March 27-31, 2000 in Morelia, Mexico
Organizing Committee: Susan Montgomery (USC), Jose Antonio de la Pena
(UNAM), Claudio Procesi (U. of Roma), and Nicolai Reshetikhin (UCB).
Quantum groups emerged from mathematical physics in mid 80's as an
algebraic structure hidden behind quantum integrable systems.
Algebraically quantum groups are Hopf algebras which are noncommutative
deformations of functions on Lie groups, or dualizing, non-commutative
deformations of universal enveloping algebras of Lie algebras. Immediately
after these structures were discovered they were used to construct new
invariants of knots and 3-manifolds.
One of the most important discoveries in representation theory in the
90's was the universal (crystal) basis discovered by Kashiwara and
Lusztig, discovered using quantum groups, and more recently, Nakagima and
others constructed representations of affine Lie algebras and
corresponding quantum groups using geometry of certain moduli spaces.
Another area where quantum groups clarified a lot the existing results
and made possible fast progress in the theory of special functions
(q-special functions). Conceptually, this direction can be regarded as
harmonic analysis on quantum groups. Yet another direction emerged from
study of the study of Hopf algebras with real structure by means of
functional analysis. This direction, is well represented in the community
of people working in C*-algebras.
Topics to be covered in the conference are as follows:
* Finite dimensional Hopf algebras
* Geometric realizations of quantized universal enveloping
algebras
* Applications of quantum groups
* Representation theory of quantum groups
This workshop will be held March 27-31, 2000 in Morelia, Mexico. A
proposal for funding has been submitted jointly to the NSF and CONACyT. It
is anticipated that there will be approximately 70 participants, half of
whom are NSF supported, and the others supported by CONACyT. Preference
will be given to recent PhD's and graduate students.
To apply for financial support: To apply for funding, send a letter
explaining your interest in the workshop together with a vita or
bibliography and a budget for travel/living expenses. If you are a
student, also solicit a letter from a faculty advisor. Those coming
from North America should apply through MSRI; those coming from Latin
America should apply through the Instituto de Matematicas, UNAM (contact
Jose Antonio de la Pena, jap@penelope.matem.unam.mx). All information
should be received by December 1, 1999.
For more information: Communications about this workshop should be
sent either by email to
qgroups@msri.org
or by regular mail to:
Quantum Groups
Mathematical Sciences Research Institute
1000 Centennial Drive
Berkeley, CA 94720-5070.
Topic #4 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: Symposium on Trends in Approximation Theory
[This information is taken from the web site:
http://www.math.vanderbilt.edu/~at/]
TRENDS IN APPROXIMATION THEORY
An International Symposium Celebrating the 60th Birthday of Larry L.
Schumaker will be held in connection with the 15th annual Shanks Lecture
at Vanderbilt University, Nashville, Tennessee on May 17-20, 2000. The
Plenary Speakers are:
Charles Chui (Stanford, USA)
Zbigniew Ciesielski (Sopot, Poland)
Ron DeVore (Columbia, USA)
Nira Dyn (Tel-Aviv, Israel)
Manfred von Golitschek (Wuerzburg, Germany)
Jacob Korevaar (Amsterdam, The Netherlands)
George G. Lorentz (Chico, USA) - Shanks Lecturer
Sergej Mikhajlovich Nikol'skii (Moscow, Russia)
Richard Varga (Kent, USA)
Contributed Talks
We invite you to contribute a talk in any area of approximation
theory and its applications. The duration of contributed talks
will depend on the number of participants and will be announced
later.
Symposium Topics
The topics of interest include, but are not limited to:
Abstract approximation
Approximation with constraints
Classical approximation
Complex approximation
Extremal problems
Interpolation and smoothing
Curves and surfaces
Multiresolution analysis
Nonlinear approximation
Orthogonal polynomials
Radial basis functions
Shift-invariant spaces
Splines
Subdivision and refinable functions
Image and signal processing
Wavelets
Proceedings
We expect to publish a proceedings containing survey papers by
the invited speakers and refereed contributed papers.
Financial Support
We are currently applying for funding to be able to partially
support the expenses of graduate students and other
mathematicians without support.
Organizing Committee
Kirill Kopotun (Vanderbilt University, USA)
Tom Lyche (University of Oslo, Norway)
Mike Neamtu (Vanderbilt University, USA)
e-mail: at@math.vanderbilt.edu
Symposium Address
Trends in Approximation Theory 2000
Department of Mathematics
Vanderbilt University
1326 Stevenson Center
Nashville, TN 37240
USA
More information is available at the Symposium Web site:
http://www.math.vanderbilt.edu/~at/
Topic #5 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Martin Muldoon
August 14-18: International Symposium on Applied Mathematics,
Dalian, China
[This is extracted from a conference poster.]
Main Sessions
- Applied Partial Differential Equations
- Applied Probability and Statistics
- Approximation Theory
- Asymptotic Analysis
- Computational Geometry
- Dynamical Systems and Fractals
- Scientific Computing
- Special Functions
- Wavelets
Plenary Speakers
- A. Jeffrey (University of Newcastle Upon Tyne, UK)
- D. Benney (MIT, USA)
- L. Gatteschi (University of Torino, Italy)
- W. Gautschi (Purdue University, USA)
- T.-T. Li (Fudan University, China)
- Q. Lin (Chinese Academy of Sciences, China)
- Z.-M. Ma (Chinese Academy of Sciences, China)
- C. A. Michelli (IBM, USA)
- R. Miura (University of British Columbia, Canada)
- B. Moodie (University of Alberta, Canada)
- O. Nevanlinna (Helsinki University of Technology, Finland)
- Z.-C. Shi (Chinese Academy of Sciences, China)
- U. Shokin (Russian Academy of Sciences, Russia)
- B. Sleeman (University of Leeds, UK)
Deadline for submitting abstracts: March 31, 2000
Local Contact Person:
Prof. Wei Wu
Department of Applied Mathematics
Dalian University of Technology
Dalian, China 116023
Email: wuweiw@dlut.edu.cn
Fax: 86-411-4708360
Scientific Committee:
R.-H. Wang (Dalian University of Technology)
R. Wong (City University of Hong Kong)
Conference Co-ordinators:
W. Wu (Dalian University of Technology)
Benny Hon (City University of Hong Kong)
Topic #6 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: OPSFA-Patras
Earlier announcements on the International Symposium on Orthogonal
Polynomials, Special Functions and Their Applications to be held in
Patras, Greece, September 20-24, 1999 have appeared in OP-SF NET 5.4 #3
and 6.1 #8. The Third Circular has been sent to those registered and
appears in the Symposium web site:
http://www.math.upatras.gr/opsfa/
Here we repeat some of the scientific information concerning the
Symposium; a much fuller picture giving local information and list of 95
contributed talks can be accessed at the web site.
The Conference is dedicated to Professor Theodore Chihara.
The plenary lectures are as follows:
1. D. Bessis, On the application of moment methods to the positive
solution of a certain non-linear elliptic and parabolic partial
differential equations in arbitrary dimensions
2. T. Chihara, 45 years of Orthogonal Polynomials: a View from the
Wings
3. J. Dehesa, Entropy-like functionals for Orthogonal Polynomials,
Asymptotics and some Physical Applications
4. A. Elbert, Some recent results on the zeros of Bessel functions and
Orthogonal Polynomials
5. N. Everitt, Orthogonal polynomials, linear differential equations
and the Kramer sampling theory
6. W. Gautschi, The use of rational functions in numerical quadrature
7. T. Kriecherbauer, A Riemann-Hilbert approach to asymptotic
questions for Orthogonal Polynomials
8. A. Kuijlaars, Asymptotics of Orthogonal Polynomials with Slowly
Decaying Weights
9. V. Sorokin, Applications of simultaneous orthogonal polynomials in
number theory, theoretical physics and dynamical systems.
Topic #7 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: Hong Kong Panel Discussion
This is a report on the Panel discussion held on Thursday, June 24, 1999,
5.00-6.30 p.m. during the International Workshop on Special Functions,
Asymptotics, Harmonic Analysis, and Mathematical Physics at the City
University of Hong Kong.
The basic account was written by Tom Koornwinder modified by input from
the participants and from Daniel Lozier, Martin Muldoon and Andre
Ronveaux.
The session was chaired by Charles Dunkl. He stressed the importance of
a discussion of important directions so that the rest of the world would
know what we were up to. This could be useful both to funding agencies
and to inform young people of directions in research. He proposed a
discussion of research perspectives for, successively:
1. Asymptotics
2. Harmonic analysis
3. Classical special functions
4. Mathematical physics
In the ensuing discussion, classical special functions were skipped as a
separate item, but they were covered by the item on harmonic analysis.
Mourad Ismail raised the issue of future meetings in the series
Fields-Toronto (1995) - CRM-Montreal (1996) - Mount Holyoke (1998) - Hong
Kong (1999) - Arizona (2000) ... with the suggestion that there should be
more than one year between meetings and that perhaps there should be a
joint meeting with the (mainly European) series on Orthogonal Polynomials.
There was unanimous agreement that the series should continue and support
for the idea of a meeting in the Netherlands in 2002 to be organized by
Tom Koornwinder, Nico Temme and Erik Koelink. Luc Vinet proposed that
there should be some kind of coordinating group for these meetings. There
was general agreement that the SIAM Activity Group on Orthogonal
Polynomials and Special Functions might provide this coordination. It was
stressed that this did not include the actual organization of meetings.
1. Asymptotics
1.1. Frank Olver
For ordinary differential equations it is time to write another
book to replace W. Wasow's "Asymptotic expansions for ordinary
differential equations", Wiley, 1965. The new book should be more
practical, giving simpler proofs, examples, error bounds (where
possible), increased regions of validity in the complex plane, and a
description of the different types of asymptotic solutions (explicit
and implicit). Some of the recent work on re-expansions of the
remainder terms (hyperasymptotics) should also be included.
For difference equations, the whole asymptotic theory needs reworking
in a much simpler and more readily applicable form than in the
classic (and almost impenetrable) papers of G. D. Birkhoff and W.J.
Trjitzinsky. Proofs can be given (by use of boundary-value type
methods) that obviate the need to pass from the set of integers on
which the solutions actually live, into the complex plane--with all
its attendant problems of analyticity. The two 1992 papers of R.
Wong and H. Li are a good beginning. Expansions should be sought in
inverse factorial series as well as, or in place of, conventional
expansions in inverse powers.
1.2. Nico Temme
For asymptotics of integrals, the theory and construction of error bounds
lags behind the corresponding results for differential equations.
1.3. Roderick Wong
He agreed with Temme. In addition to the problem of error bounds, for
asymptotics of integrals, regions of validity are usually smaller than the
corresponding ones that can be established by using the differential
equation theory. This situation needs to be improved. For asymptotic
solutions of difference equations, Airy-type expansions should be
considered. He referred to a paper (by Costin and Costin) in SIAM J Math
Anal. In singular perturbations, derivations of asymptotics are mostly
formal; rigorous proofs may give more insight. For the nonlinear
Klein-Gordon equation, even to show that the solution is uniformly bounded
for all time is not a simple matter. For specific problems even with two
terms, it is difficult to show that the remainder is of the right order.
1.4. Robert O'Malley
Parabolic p.d.e.'s sometimes have travelling wave solutions (like
hyperbolic tangent in Burgers' equation). To obtain such solutions one has
to go back to solutions of very special o.d.e.'s and their asymptotics.
Dumb computing won't work there. Applied people have to learn about the
powerful tools of solutions by special functions and their asymptotics.
1.5. Eric Opdam
People in this workshop include, on the one hand, researchers with very
detailed knowledge of one-variable theory and, on the other hand, workers
in higher rank theory (i.e. more variable theory associated with root
systems). They should join forces in order to do asymptotics in the higher
rank case. Olver commented that in asymptotics the difficulty goes up by
an order of magnitude as each new variable is added. Moreover, because of
the increasing complexity a point of diminishing returns is soon reached.
Koelink responded that the symmetry available in the higher rank case may
help in doing asymptotics.
2. Harmonic analysis
2.1. George Gasper
A big problem is that of convolution structures.
He suggested a study of nonnegative kernels, as done for q-Racah
polynomials in the paper
G. Gasper and M. Rahman, "Nonnegative kernels in product formulas
for q-Racah polynomials", J. Math. Anal. Appl. 95 (1983), 304-318
They were able to handle only the case with symmetry; the non-symmetric
case is still open. Many other cases (of convolution structures) are
still unknown.
Dunkl remarked that all positivity results are important.
2.2. Erik Koelink
Be aware of possible applications of special functions in signal analysis,
stochastic processes, financial mathematics.
Furthermore, a lot of inspiration for further work can be obtained from
the work by Percy Deift on Riemann-Hilbert problems and random matrix theory,
see for instance
P. Deift, "Orthogonal polynomials and random matrices: a
Riemann-Hilbert approach", Courant Lecture Notes in Math., Vol. 3,
Courant Institute of Math. Sciences, 1999.
See also the topics covered in this fall's special semester on
Foundations of Computational Mathematics in Hong Kong
(http://www.damtp.cam.ac.uk/user/na/FoCM/HK.html),
in particular the workshop on Minimal energy problems
(http://www.math.usf.edu/FoCM99/).
Finally, KZ equations and elliptic hypergeometric functions are promising
fields.
2.3. Richard Askey
Terwilliger's results on Leonard pairs (see Terwilliger's lecture in this
workshop) gives a new approach to Leonard's characterization of q-Racah
polynomials and some related polynomials in
D.A. Leonard, "Orthogonal polynomials, duality and association schemes",
SIAM J. Math. Anal. 13 (1982), 656-663.
Terwilliger's approach works for any field. What are the implications
for special functions for other fields than the reals?
Next, Askey suggests finding an integral equation for the
so-called Barnes G-function, which satisfies G(x+1)=gamma(x)*G(x), G(1)=1.
There is a problem on this function in Whittaker and Watson with a reference
to late 19th century work by a Russian. Barnes has a number of papers
on this function, in the early part of this century.
There might be an infinite dimensional integral
which represents this function, based on the form Selberg's integral
takes. The G-function shows up in a number of places.
Andrew Lenard used it in a paper (J. Math. Phys. 5 (1964), 930-943) on the
strong Szego limit theorem. Szego pointed out to him how some of
his results could be stated using the G-function. It also shows up
in K-theory.
Finally, for 9j-symbols, a representation as a double sum should be found.
Such a representation can be expected, since they have a two variable
orthogonality.
2.4. Christian Berg
1. Concerning the indeterminate moment problem one should try to find a
closer relationship between the growth rate of the coefficients of the
three terms recurrence relation for the orthonormal polynomials and the
growth properties of the entire functions in the Nevanlinna matrix. In
particular one should relate the order of these functions to the growth
rate of the coefficients.
In a special case of birth and death rates being a specific polynomial in
n of degree four, it turned out that the entire functions have order 1/4,
see C. Berg and G. Valent, "The Nevanlinna parametrization for some
indeterminate Stieltjes moment problems associated with birth and death
processes", Meth. Appl. Anal 1 (1994), 169-209. In a recent manuscript by
G. Valent,"Indeterminate moment problems and a conjecture on the growth of
the entire functions of the Nevanlinna parametrization", there is an
example of birth and death rates being polynomials of degree 3 and the
order of the entire functions are 1/3. The paper also formulates a
conjecture.
2. It was proved by C. Berg and W. Thill, "Rotation invariant moment
problems", Acta Math. 167 (1991), 207-227, that a moment problem in
R^n, n > 1, can have a unique solution mu (i.e. mu is a determinate
measure on R^n) and yet the polynomials are not dense in the Hilbert space
L^2(R^n,mu). Apart from the rotationally invariant case no criterion
seems to be known for this phenomenon to happen, and this kind of question
needs further study.
2.5. Eric Opdam
Special function theory reflects deep properties in group theory,
mathematical physics and number theory. For advances in special functions
one should better understand these three fields. In mathematical
physics the Calogero-Moser system is a deformation of the boson gas. Find
correlation functions for the Calogero-Moser system as generalizations of
such functions for the boson gas. Macdonald theory corresponds to the
boson gas with periodic constraints. In getting rid of these constraints
one would arrive at Macdonald functions for the noncompact case (see
lectures by Koelink and Stokman at this workshop for the rank one case),
which might be studied from the point of view of double affine Hecke
algebras. In representation theory one should not restrict oneself to the
spherical case. Askey added that we need to get past page 2 of Zygmund
and start solving hard problems with real applications.
2.6. Yuan Xu
Most of the L^p theory for orthogonal polynomials in several variables
is not yet understood.
There is the question of finding explicit formula for orthogonal
polynomials associated to a weight function that is invariant under an
octahedral group. For example, Dunkl's h-harmonics associated to the type
B weight. Such a basis may be useful in studying cubature formulae
(numerical integration formulae). There is a possible connection between
common zeros of invariant orthogonal polynomials for the weight function
(x1^2 - x2^2)^2(x2^2 - x3^2)^2(x3^2 - x1^2)^2 on the sphere S^2 and a
family of cubature formulae on S^2 conjectured by V. I. Lebedev.
2.7. Dennis Stanton
Macdonald's conjecture about the positivity of the coefficients in
the polynomial expansion of the Macdonald-Kostka coefficients and
the Garsia-Haiman n factorial conjecture
are at present the most important conjectures in algebraic combinatorics,
see for instance
A.M. Garsia and M. Haiman, Some natural bigraded $S_n$-modules and
$q,t$-Kostka coefficients. The Foata Festschrift. Electron. J. Combin.
3 (1996), no. 2, Research Paper 24,
http://www.combinatorics.org/Volume_3/Abstracts/v3i2r24.html
Furthermore, problems associated with graph spectra for finite matrix groups
(A. Terras, lecture at this workshop) are very important.
2.8. Audrey Terras
Graph spectra could lead to some interesting special functions.
2.9. Charles Dunkl
1) I suggest the study of orthogonal polynomials (special functions,
transforms) associated with the non-crystallographic reflection groups,
that is, H3 and H4; of icosahedral type. These groups are related to
quasicrystals; an area of research in both physics and mathematics.
2) Alberto Gruenbaum (referring to his work with Duistermaat on bispectral
problems) suggested to me earlier, the idea of finding bispectral
differential-difference operators (eigenfunctions of one operator satisfy
another d-d equation with respect to the eigenvalue). Alberto and I worked
out a small example (a new way of looking at a known situation) which
involved a perturbed one-variable differential-difference operator related
to the group Z2. Thus I speculate there may be perturbations for a limited
set of parameter values (more speculation: those called singular by de
Jeu, Opdam and myself).
3) I speculate that the phenomenon called super-integrability (e.g.,
Konstein) may apply to algebras of differential-difference operators
with integer parameter values.
2.11. Mourad Ismail
Study asymptotics of
$$
P_n(x):= c_n \int_a^b \ldots \int_a^b \prod_{j=1}^n (x-\lambda_j)
\prod_{1\le i
Subject: Errors in Andrews-Askey-Roy
Cambridge University Press is likely to put out a paperback version of
"Special Functions" by Andrews, Askey and Roy [1]. It is possible to make
corrections, so George, Ranjan and I would appreciate learning of all of
those you have found. These can be sent to me, or better, to all three of
us. Our addresses are:
askey@math.wisc.edu
andrews@math.psu.edu
royr@beloit.edu
Lauren Cowles said she would like these by October 15, so send what you
have now and others later. Feel free to share this message with others
who might know of mistakes.
[1] Andrews, George E., Askey, Richard and Roy, Ranjan: "Special
Functions", Encyclopedia of Mathematics and its Applications 71, 1998, c.
560 pp., Hardback, 0-521-62321-9, $85.00
[Editor's Note: This book was announced in OP-SF NET 6.1, Topic 11. We
hope to feature a review of it in the next issue.]
Topic #9 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Jan Felipe van Diejen
Subject: Proceedings "Algebraic Methods and q-Special Functions"
PROCEEDINGS OF THE CRM WORKSHOP
"ALGEBRAIC METHODS AND q-SPECIAL FUNCTIONS",
MONTREAL, 1996.
In May 1996, the Centre de Recherches Mathematiques at Montreal organized
the workshop "Algebraic Methods and q-Special Functions". The idea of
this workshop was to bring together a diverse group of people working in
the areas of combinatorics, special functions, orthogonal polynomials and
representation-theoretic methods, with the purpose of getting some kind of
an overview of the current developments in the field. Much of the
reported research progress was inspired by the seminal contributions of R.
A. Askey and colleagues on one-variable basic hypergeometric series and of
I. G. Macdonald on multivariate orthogonal polynomials related to root
systems.
Recently, the proceedings of this workshop appeared in the CRM Proceedings
and Lecture Notes Series published by the AMS:
"Algebraic Methods and q-Special Functions",
Eds.: Jan Felipe van Diejen (Universidad de Chile, Santiago, Chile)
and Luc Vinet (Université de Montreal, Quebec, PQ, Canada),
CRM Proceedings and Lecture Notes 22, American Mathematical Society,
Providence, R.I., 1999.
ISBN: 0-8218-2026-5
Paging: 276 pp.
Binding: Softcover
List Price: $79
Institutional Member Price: $63
Individual Member Price: $47
Order Code: CRMP/22
Contents
F. Bergeron and A. M. Garsia -- Science fiction and Macdonald's
polynomials
R. Chouikha -- On the expansion of elliptic functions and applications
D. V. Chudnovsky and G. V. Chudnovsky -- Generalized hypergeometric
functions - Classification of identities and explicit rational
approximations
W. S. Chung, E. G. Kalnins, and W. Miller, Jr. -- Tensor products of
q-superalgebras and q-series identities. I
J. F. van Diejen and J. V. Stokman -- q-Racah polynomials for BC type
root systems
C. F. Dunkl -- Intertwining operators of type B_N
R. Floreanini, J. LeTourneux, and L. Vinet -- Symmetries and continuous
q-orthogonal polynomials
P. G. A. Floris -- Addition theorems for spherical polynomials on a
family of quantum spheres
F. A. Grunbaum and L. Haine -- On a q-analogue of the string equation
and a generalization of the classical orthogonal polynomials
M. E. H. Ismail, D. R. Masson, and S. K. Suslov -- The q-Bessel function
on a q-quadratic grid
D. Kim and D. Stanton -- Three statistics on lattice paths
A. N. Kirillov -- Quantum Grothendieck polynomials
A. N. Kirillov and M. Noumi -- q-difference raising operators for
Macdonald polynomials and the integrality of transition
coefficients
B. A. Kupershmidt -- Great powers of q-calculus
V. Spiridonov -- q-special functions: Differential-difference equations,
roots of unity, and all that
A. Strasburger -- On algebras of creation and annihilation operators
Further information on these proceedings can be obtained via the AMS
Bookstore WEB page:
http://www.ams.org/cgi-bin/bookstore/bookpromo/crmpseries
Topic #10 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Wolfram Koepf
Subkect: Proceedings "Applications and Computation of Orthogonal
Polynomials"
This is an announcement from the publisher (Birkhauser):
Gautschi, W, Golub, G. H. and Opfer, G.: Applications and Computation of
Orthogonal Polynomials. (Conference at the Mathematical Research
Institute Oberwolfach, Germany, March 22-28, 1998)
1999. 288 pages. Hardcover
sFr. 148.ű / DM 178.ű / oS 1300.
ISBN 3-7643-6137-9
This volume contains a collection of papers dealing with applications of
orthogonal polynomials and methods for their computation.
The applications address problems in applied mathematics as well as
problems in engineering and the sciences. Prominent among the former are
least-squares approximations, Gauss and related quadrature, iterative
methods in linear algebra, the detection of singularities, and integral
equations. Applications of the latter kind include the use of wavelets
in medical diagnostics and the relevance of orthogonal polynomials in
optimal control, dynamical systems, and gas dynamics. Computational
methods relate to numerical and symbolic computation and include, in
particular, matrix interpretation and convergence, perturbation, and
stability analyses of relevant algorithms. Generalizations of orthogonal
polynomials are also considered, for example, s-orthogonal, matrix- and
tensor-valued, Muntz type, and complex orthogonal polynomials.
Topic #11 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Walter Van Assche
Subject: Proceedings "Continued Fractions: from analytic number theory to
constructive approximation"
"Continued Fractions: from analytic number theory to constructive
approximation" a volume in honor of L.J. Lange.
University of Missouri, Columbia, May 20-23, 1998.
Bruce C. Berndt, Fritz Gesztesy, editors
in "Contemporary Mathematics" volume 236, Amer. Math. Soc., 1999
The volume contains the following articles:
R. Askey: Continued fractions and orthogonal polynomials
B.C. Berndt, Y.-S. Choi, S.-Y. Kang: The problems submitted by
Ramanujan to the Journal of the Indian Mathematical Society
B. Bojabov, A. Sri Ranga: Some examples of moment preserving
approximation
C.F. Bracciali: Relations between certain symmetric strong Stieltjes
distributions
A. Bultheel, C. Diaz-Mendoza, P. Gonzalez-Vera, R. Orive: Estimates
of the rate of convergence for certain quadrature formulas on the
half line
A. Bultheel, P. Gonzalez-Vera: Wavelets by orthogonal rational kernels
H.H. Chan, V. Tan: On the explicit evaluations of the Rogers-Ramanujan
continued fraction
D. Chelst: Absence of phase transitions in modified two-component
plasmas: the analytic theory of continued fractions in statistical
mechanics
M.E.H. Ismail, D.R. Masson: Some continued fractions related to
elliptic functions
W.B. Jones, G. Shen: Asymptotics of Stieltjes continued fraction
coefficients and applications to Whittaker functions
L.J. Lange: A generalzation of Van Vleck's theorem and more on complex
continued fractions
X. Li: Convergence of interpolation Laurent polynomials on an annulus
L. Lorentzen: Convergence criteria for continued fractions
K(a_n/1) based on value sets
O. Njastad: Strong Stieltjes moment problems
F. Peherstorfer, R. Steinbauer: Weak asymptotics of orthogonal
polynomials on thje support of the measure of orthogonality and
considerations on functions of the second kind
S. Perrine: Trees of approximation constants
I. Rodnianski: Continued fractions and Schrodinger evolution
W. Van Assche: Multiple orthogonal polynomials, irrationality and
transcendence
A.J. van der Poorten: Reduction of continued fractions of formal power
series
H. Waadeland: Some observations in frequency analysis
F. Wielonsky: Some properties of Hermite-Pade approximants to e^z
Topic #12 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: Askey featured in SIAM News article
The July/August 1999 (Volume 32, Number 6) issue of SIAM News carries an
article "Askey, Rokhlin elected to NAS" on Richard Askey and Vladimir
Rokhlin on their election to the US National Academy of Sciences. The
material on Askey, written by Walter Van Assche, Vice-Chair of our
Activity Group, gives a succinct account of Askey's many-sided
contributions and leadership. The web site
http://www.siam.org/siamnews/index.htm
gives access to issues of SIAM News. The July/August issue has not been
posted at the time of this writing.
Topic #13 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Daniel W Lozier
Subject: NSF Funding Received for NIST Digital Library Project
The US National Science Foundation recently awarded the National Institute
of Standards and Technology $1.3 million for the NIST Digital Library of
Mathematical Functions. The money will be used to support contracts with
expert non-NIST authors and validators to write chapters for the DLMF.
The DLMF was conceived as the successor for the Handbook of Mathematical
Functions, edited by Abramowitz and Stegun and published in 1964 by the
National Bureau of Standards (now known as NIST). The style and approach
follows Abramowitz and Stegun, but the new reference work will be
disseminated from a Web site now under construction at NIST as well as in
book form (probably with included CD-ROM). The project was introduced at
the 1997 SIAM annual meeting in the Minisymposium on Handbooks for Special
Functions and the World Wide Web, organized by our activity group.
Articles appeared in OPSF-Net Volume 4, Number 5, September 15, 1997, and
in the March 1998 issue of SIAM News.
The NSF award is under the Knowledge and Distributed Intelligence program,
which seeks to foster the development of advanced research tools using the
emerging capabilities of information technology. The DLMF will satisfy
diverse user requirements such as simple lookup, complex search and
retrieval, formula validation, interactive visualization, and pointers to
software and evaluated numerical methodology. The award period is three
years.
The current table of contents includes 39 chapters. Most of these address
classical special functions and orthogonal polynomials but included also
are chapters on relevant methodology (e.g. asymptotic approximations,
numerical methods, computer algebra) and recently developed topics (e.g.
generalized and basic hypergeometric functions, discrete orthogonal
polynomials, Painleve transcendents). The project is being supervised by
an editorial board consisting of four NIST editors and ten non-NIST
associate editors. For further details, see the project Web site:
http://math.nist.gov/DigitalMathLib.
Topic #14 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Allison Bogardo
Subject: SIAM Student Paper Prizes
SIAM Student Paper Prizes
The annual SIAM Student Paper Prizes will be awarded during the 2000
SIAM Annual Meeting, July 10-14, at the Westin Rio Mar Beach Resort
in Rio Grande, Puerto Rico.
If you are a student or know of a student who would like to take part
in the competition, here are the details:
The authors of the three best papers in applied and computational
mathematics written by students and submitted to SIAM will receive a
$1,000 cash prize and a framed calligraphed certificate as well as
gratis registration for the meeting. There is no provision for
travel expenses associated with the prize.
Papers must be singly authored and not previously published or
submitted for publication to be eligible for consideration. To
qualify, authors must be students in good standing who have not
received their PhDs at the time of submission.
In submitting their work for publication, authors are asked to
consider SIAM journals. However, student paper prize winners are not
guaranteed publication in any SIAM journal; all papers submitted to
SIAM journals are subject to the same refereeing process and
standards.
Submissions must be received in the SIAM office before February 15,
2000.
Submissions, which must be in English, can be sent by regular mail or
fax. Each submission must include (1) an extended abstract NOT
LONGER THAN 5 PAGES (including bibliography); (2) the complete paper,
which will be used solely for clarification of any questions; (3) a
statement by the student's faculty advisor that the paper has been
prepared by the author indicated and that the author is a student in
good standing; (4) a letter by the student's faculty advisor
describing and evaluating the paper's contribution; and (5) a short
biography of the student.
Submissions will be judged on originality, significance, and quality
of exposition.
The winners will be notified by April 15, 2000.
Please direct your submission and any questions you may have to
A. Bogardo at SIAM, 3600 University City Science Center,
Philadelphia, PA 19104-2688; telephone (215) 382-9800; e-mail to
bogardo@siam.org.
Topic #15 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Allison Bogardo
Subject: SIAM Student Travel Awards
SIAM Student Travel Awards
for 2000 Conferences
During 2000, SIAM will make a number of awards for $300 to support
student travel to each of the following SIAM conferences:
Eleventh ACM-SIAM Symposium on Discrete Algorithms, San Francisco,
California, January 9-11.
Eighth International Conference on Numerical Combustion, Amelia
Island, Florida, March 5-8.
Third SIAM Conference on Mathematical Aspects of Materials Science,
Philadelphia, Pennsylvania, May 21-24.
Tenth SIAM Conference on Discrete Mathematics (SIAG/DM), Minneapolis,
Minnesota, June 12-15.
2000 SIAM Annual Meeting, Rio Grande, Puerto Rico, July 10-14.
Pacific Rim Dynamical Systems Conference (SIAG/DS), Maui, Hawaii,
August 10-12
First SIAM Conference on Computational Science and Engineering,
Washington, DC, September 21-23.
Seventh SIAM Conference on Applied Linear Algebra, Raleigh, North
Carolina, October 23-26.
The awards are to be made from the SIAM Student Travel Fund, created
in 1991 and maintained through book royalties donated by generous SIAM
authors.
Any full-time student in good standing is eligible to receive an award
plus gratis meeting registration. Top priority will be given to
students presenting papers at the meeting, with second priority to
students who are co-authors of papers to be presented at the meetings.
Only students traveling more than 100 miles to the meetings are
eligible for the awards.
An application for a travel award must include:
(1) A letter from the student describing his/her academic standing and
interests, his/her expected graduation date and degree, advisor's
name, and, if available, a URL for a working Web page.
(2) A one-page vita that includes the student's research interests,
projects, and papers published.
(3) A detailed letter from the student's faculty advisor indicating
why the student is deserving of receiving a travel award and any
special circumstances.
(4) If applicable, the title(s) of the paper(s) to be presented
(co-authored) by the student at the meeting.
Applications should be sent to the SIAM office (Attention: SIAM
Student Travel Awards), 3600 University City Science Center,
Philadelphia, PA 19104-2688. Students also may apply by e-mail to
bogardo@siam.org or by fax to 215-386-7999.
Complete applications must be received at the SIAM office no later
than TWO MONTHS before the first day of the meeting for which support
is requested.
Winners will be notified FIVE WEEKS before the first day of the
meeting. Checks for the awards will be given to the student awardees
when they arrive at the given meeting and pick up their registration
packet at the SIAM Registration Desk.
Topic #16 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: Allison Bogardo
Subject: Call for Nominations/Polya Prize
CALL FOR NOMINATIONS
for
GEORGE POLYA PRIZE
The Polya Prize
---------------
SIAM will present the award at the 2000 SIAM Annual Meeting in Rio
Grande, Puerto Rico, July 10-14. The award honors the memory of
George Polya and will be given for a notable contribution in
combinatorial theory.
Eligibility
-----------
There are no restrictions.
Description of Award
--------------------
The award will consist of an engraved medal and a $20,000 cash prize.
Travel to the SIAM meeting to receive the award will be paid by the
prize fund.
Nominations
-----------
A letter of nomination, including a description of achievement(s),
should be sent by December 31, 1999, to:
Professor Jeffry N. Kahn
Chair, Polya Prize Selection Committee
c/o A. G. Bogardo
SIAM
3600 University City Science Center
Philadelphia, PA 19104-2688
E-mail: bogardo@siam.org
FAX: 215-386-7999
Selection Committee
-------------------
The members of the selection committee for the award are Jeffry N.
Kahn (Rutgers University), chair; Louis J. Billera (Cornell
University); Joel Spencer (Courant Institute, New York University);
and Richard P. Stanley (Massachusetts Institute of Technology).
Topic #17 ------------ OP-SF NET 6.5 ----------- September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: OP-SF preprints in xxx archive
The following preprints related to the field of orthogonal
polynomials and special functions were recently posted or cross-listed to
one of the subcategories of the xxx archives. See:
http://front.math.ucdavis.edu/math.CA
http://front.math.ucdavis.edu/math.CO
http://front.math.ucdavis.edu/math.QA
http://xxx.lanl.gov/archive/solv-int
math.CA/9908163
Title: Inversion formulas involving orthogonal polynomials and some of
their applications
Author: Roelof Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45
Comments: 15 pages, submitted for publication in the Proceedings of the
International Workshop on Special Functions (Asymptotics, Harmonic
Analysis and Mathematical Physics), City University of Hong Kong, Kowloon,
Hong Kong, June 21-25, 1999
Abstract: We derive inversion formulas involving orthogonal polynomials
which can be used to find coefficients of differential equations satisfied
by certain generalizations of the classical orthogonal polynomials. As an
example we consider special symmetric generalizations of the Hermite
polynomials.
From: Roelof Koekoek
math.CA/9908162
Title: Differential equations for generalized Jacobi polynomials
Authors: J. Koekoek, R. Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Report number: 98-42
Comments: 33 pages, submitted for publication
Abstract: We look for spectral type differential equations satisfied by
the generalized Jacobi polynomials, which are orthogonal on the interval
[-1,1] with respect to a weight function consisting of the classical
Jacobi weight function together with two point masses at the endpoints of
the interval of orthogonality.
We show that such a differential equation is uniquely determined and
we give explicit representations for the coefficients.
In case of nonzero mass points the order of this differential
equation is infinite, except for nonnegative integer values of (one of)
the parameters. Otherwise, the finite order is explicitly given in terms of
the parameters.
From: Roelof Koekoek
math.CA/9908148
Title: The Jacobi inversion formula
Authors: J. Koekoek, R. Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Journal reference: Complex Variables 39 (1999) 1-18
Comments: 15 pages
Abstract: We look for spectral type differential equations satisfied by
the generalized Jacobi polynomials which are orthogonal on the interval
[-1,1] with respect to a weight function consisting of the classical
Jacobi weight function together with point masses at the endpoints of the
interval of orthogonality.
In order to find explicit formulas for the coefficients of these
differential equations we have to solve systems of equations involving
derivatives of the classical Jacobi polynomials. These systems of
equations have a unique solution which can be given explicitly in terms of
Jacobi polynomials. This is a consequence of the Jacobi inversion formula
which is proved in this paper.
From: Roelof Koekoek
math.CA/9908147
Title: Finding differential equations for symmetric generalized
ultraspherical polynomials by using inversion methods
Authors: J. Koekoek, R. Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Comments: 10 pages, published in Proceedings of the International Workshop
on Orthogonal Polynomials in Mathematical Physics (Leganes, 1996),
Universidad Carlos III Madrid (Leganes, 1997), 103-111. See also :
http://dulcinea.uc3m.es/users/workshop/proceedings.html
Abstract: We find all spectral type differential equations satisfied by
the symmetric generalized ultraspherical polynomials which are orthogonal
on the interval [-1,1] with respect to the classical symmetric weight
function for the Jacobi polynomials together with equal mass points at
both ends of the interval of orthogonality.
In order to find explicit formulas for the coefficients of these
differential equations we have to solve systems of equations involving
derivatives of the classical Jacobi polynomials. These systems of
equations have a unique solution which is given explicitly. This is a
consequence of the Jacobi inversion formula.
From: Roelof Koekoek
math.CA/9908146
Title: Inversion methods for finding differential equations for
generalized Jacobi polynomials
Authors: J. Koekoek, R. Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Report number: 96-105
Comments: 23 pages
Abstract: We look for differential equations satisfied by the generalized
Jacobi polynomials which are orthogonal on the interval [-1,1] with
respect to a weight function consisting of the classical Jacobi weight
function together with point masses at the endpoints of the interval of
orthogonality.
In order to find explicit formulas for the coefficients of these
differential equations we have to solve systems of equations involving
derivatives of the classical Jacobi polynomials.
We show that these systems of equations have a unique solution which
is given explicitly. This is a consequence of the Jacobi inversion formula
which is proved in this report.
From: Roelof Koekoek
math.CA/9908145
Title: On differential equations for Sobolev-type Laguerre polynomials
Authors: J. Koekoek, R. Koekoek, H. Bavinck
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Journal reference: Trans. Amer. Math. Soc. 350 (1998) 347-393
Report number: 95-79
Comments: 45 pages
Abstract: We obtain all spectral type differential equations satisfied by
the Sobolev-type Laguerre polynomials. This generalizes the results found
in 1990 by the first and second author in the case of the generalized
Laguerre polynomials defined by T.H. Koornwinder in 1984.
From: Roelof Koekoek
math.CA/9908144
Title: On a difference equation for generalizations of Charlier
polynomials
Authors: Herman Bavinck, Roelof Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 39A10 (Secondary)
Journal reference: J. Approx. Theory 81 (1995) 195-206
Report number: 92-101
Comments: 11 pages
Abstract: In this paper we obtain a set of polynomials which are
orthogonal with respect to the classical discrete weight function of the
Charlier polynomials at which an extra point mass at x=0 is added. We
construct a difference operator of infinite order for which these new
discrete orthogonal polynomials are eigenfunctions.
From: Roelof Koekoek
math.CA/9908143
Title: Differential equations for symmetric generalized ultraspherical
polynomials
Author: Roelof Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Journal reference: Trans. Amer. Math. Soc. 345 (1994) 47-72
Report number: 92-08
Comments: 25 pages
Abstract: We look for differential equations satisfied by the generalized
Jacobi polynomials which are orthogonal on the interval [-1,1] with
respect to the classical weight function for the Jacobi polynomials
together with point masses at both endpoints. In the special symmetric
case that both parameters are equal and also the two point masses are
equal we find all differential equations of spectral type satisfied by
these symmetric generalized ultraspherical polynomials. We show that if
the point masses are positive only for nonnegative integer values of the
parameter there exists exactly one differential equation of spectral type
which is of finite order. By using quadratic transformations we also
obtain differential equations for some related sets of generalized Jacobi
polynomials. In these cases we find finite order differential equations
even though one of the parameters is not equal to an integer.
From: Roelof Koekoek
math.CA/9908142
Title: The search for differential equations for certain sets of
orthogonal polynomials
Author: Roelof Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Journal reference: J. Comput. Appl. Math. 49 (1993) 111-119
Comments: 10 pages, published in Proceedings of the Seventh Symposium on
Orthogonal Polynomials and Applications (VII SPOA), Granada 1991
Abstract: We look for spectral type differential equations for the
generalized Jacobi polynomials and for the Sobolev-Laguerre polynomials.
We use a method involving computer algebra packages like Maple and
Mathematica and we will give some preliminary results.
From: Roelof Koekoek
math.CA/9908141
Title: The search for differential equations for orthogonal polynomials by
using computers
Author: Roelof Koekoek
Categories: CA Classical Analysis
Math Subject Class: 33C45 (Primary) 34A35 (Secondary)
Report number: 91-55
Comments: 24 pages
Abstract: We look for spectral type differential equations for the
generalized Jacobi polynomials found by T.H. Koornwinder in 1984 and for
the Sobolev-Laguerre polynomials. We introduce a method which makes use of
computer algebra packages like Maple and Mathematica and we will give some
preliminary results.
From: Roelof Koekoek
math.CA/9908140
Title: A note on the q-derivative operator
Authors: J. Koekoek, R. Koekoek
Categories: CA Classical Analysis
Math Subject Class: 39A05 (Primary) 26A24 (Secondary)
Journal reference: J. Math. Anal. Appl. 176 (1993) 627-634
Comments: 7 pages
Abstract: We prove a formula for the nth power of the q-derivative
operator at x=0 for every function whose nth derivative at x=0 exists.. We
give a proof in both the real variable and the complex variable case.
From: Roelof Koekoek
math.CA/9907110
Title: Small eigenvalues of large Hankel matrices: The indeterminate case
Authors: Christian Berg, Yang Chen, Mourad E. H. Ismail
Categories: CA Classical Analysis
Math Subject Class: 42C05, 33C25, 31A15
Comments: 14 pages, 1 figure
Abstract: In this paper we characterise the indeterminate case by the
eigenvalues of the Hankel matrices being bounded below by a strictly
positive constant. An explicit lower bound is given in terms of the
orthonormal polynomials and we find expressions for this lower bound in a
number of indeterminate moment problems.
From: Yang Chen
math.QA/9907061
Title: The elliptic gamma function and SL(3,Z) x Z^3
Authors: Giovanni Felder (ETH Zurich), Alexander Varchenko (UNC Chapel
Hill)
Categories: QA Quantum Algebra (CA Classical Analysis)
Comments: 27 pages, LaTeX References added, minor corrections
Abstract: The elliptic gamma function is a generalization of the Euler
gamma function and is associated to an elliptic curve. Its trigonometric
and rational degenerations are the Jackson q-gamma function and the Euler
gamma function, respectively. The elliptic gamma function appears in
Baxter's formula for the free energy of the eight-vertex model and in the
hypergeometric solutions of the elliptic qKZB equations. In this paper,
the properties of this function are studied. In particular we show that
elliptic gamma functions are generalizations of automorphic forms of
G=SL(3,Z) x Z^3 associated to a non-trivial class in H^3(G,Z).
From: Giovanni Felder
math.CO/9908131
Title: Umbral presentations for polynomial sequences
Author: Brian D. Taylor
Categories: CO Combinatorics
Math Subject Class: 05A40
Comments: 19 pages
Abstract: Using random variables as motivation, this paper presents an
exposition of the formalisms developed by Rota and Taylor for the
classical umbral calculus. A variety of examples are presented,
culminating in several descriptions of sequences of binomial type in terms
of umbral polynomials.
From: Brian D. Taylor
math.AG/9908065
Title: Periods of mirrors and multiple zeta values
Author: Michael E. Hoffman (U.S. Naval Academy)
Categories: AG Algebraic Geometry (CO Combinatorics)
Math Subject Class: 14J32 (Primary); 05E05 (Secondary)
Comments: 3 pages
Abstract: In a recent paper, A. Libgober showed that the multiplicative
sequence {Q_i(c_1,...,c_i)} of Chern classes corresponding to the power
series Q(z)=1/Gamma(1+z) appears in a relation between the Chern classes
of certain Calabi-Yau manifolds and the periods of their mirrors. We show
that the polynomials Q_i can be expressed in terms of multiple zeta
values.
From: Michael E. Hoffman
math.QA/9908067
Title: A Quantum Field Theoretical Representation of Euler-Zagier Sums
Authors: Uwe Muller (Mainz Univ.), Christian Schubert (LAPTH
Annecy-le-Vieux)
Categories: QA Quantum Algebra (NT Number Theory; CO Combinatorics)
Report number: LAPTH-728/99, MZ-TH/99-35
Comments: Standard latex, 28 pages, 11 figures
Abstract: We establish a novel representation of arbitrary Euler-Zagier
sums in terms of weighted vacuum graphs. This representation uses a toy
quantum field theory with infinitely many propagators and interaction
vertices. The propagators involve Bernoulli polynomials and Clausen
functions to arbitrary orders. The Feynman integrals of this model can be
decomposed in terms of a vertex algebra whose structure we investigate.
We derive a large class of relations between multiple zeta values, of
arbitrary lengths and weights, using only a certain set of graphical
manipulations on Feynman diagrams. Further uses and possible
generalizations of the model are pointed out.
From: Christian Schubert
math.CO/9907183
Title: On Multi-color partitions and the generalized Rogers-Ramanujan
identities
Authors: Naihuan Jing, Kailash Misra, Carla Savage
Categories: CO Combinatorics (QA Quantum Algebra)
Comments: Latex2e
Abstract: Basil Gordon, in the sixties, and George Andrews, in the
seventies, generalized the Rogers-Ramanujan identities to higher moduli.
These identities arise in many areas of mathematics and mathematical
physics. One of these areas is representation theory of the infinite
dimensional Lie algebra, where various known interpretations of these
identities have led to interesting applications. Motivated by their
connections with Lie algebra representation theory, we give a new
interpretation of the product side of generalized Rogers-Ramanujan
identities in terms of multi-color partitions.
From: jing@math.ncsu.edu
math.CO/9907029
Title: A q-analogue of a formula of Hernandez obtained by inverting a
result of Dilcher
Author: Helmut Prodinger
Categories: CO Combinatorics
Math Subject Class: 05A10
Abstract: We prove a q-analogue of the formula $ \sum_{1\le k\le n} \binom
nk(-1)^{k-1}\sum_{1\le i_1\le i_2\le... \le i_m=k}\frac1{i_1i_2... i_m} =
\sum_{1\le k\le n}\frac{1}{k^m} $ by inverting a formula due to Dilcher.
From: Helmut Prodinger
Paper: math.CA/9909025
Title: A q-analogue of convolution on the line
Authors: G. Carnovale (Universite Cergy-Pontoise), T.H. Koornwinder
(Universiteit van Amsterdam)
Comments: 24 pages
Report-no: Report 99-12, Math. Preprint Series, Fac. WINS, Univ. of
Amsterdam
Subj-class: Classical Analysis; Quantum Algebra
MSC-class: 33D80, 33D15, 42A85 (primary), 17B37 (secondary)
Abstract: In this paper we study a q-analogue of the convolution product
on the line in detail. A convolution product on the braided line was
defined algebraically by Kempf and Majid. We adapt their definition in
order to give an analytic definition for the q-convolution and we study
convergence extensively. Since the braided line is commutative as an
algebra, all results can be viewed both as results in classical q-analysis
and in braided algebra. We define various classes of functions on which
the convolution is well-defined and we show that they are algebras under
the defined product. One particularly nice family of algebras, a
decreasing chain depending on a parameter running through (0,1], turns out
to have 1/2 as the critical parameter value above which the algebras are
commutative. Moreover, the commutative algebras in this family are
precisely the algebras in which each function is determined by its
q-moments. We also treat the relationship between q-convolution and
q-Fourier transform. Finally, in the Appendix, we show an equivalence
between the existence of an analytic continuation of a function defined on
a q-lattice, and the behaviour of its q-derivatives.
From: Tom H. Koornwinder
solv-int/9908002
Title: Quantum Backlund transformation for the integrable DST model
Authors: V.B. Kuznetsov, M. Salerno and E.K. Sklyanin
Comments: 24 pages, LaTeX2e
Report-no: LPENSL-TH-16/99
Abstract: For the integrable case of the discrete self-trapping (DST)
model we construct a Backlund transformation. The dual Lax matrix and the
corresponding dual Backlund transformation are also found and studied. The
quantum analog of the Backlund transformation (Q-operator) is constructed
as the trace of a monodromy matrix with an infinite-dimensional auxiliary
space. We present the Q-operator as an explicit integral operator as well
as describe its action on the monomial basis. As a result we obtain a
family of integral equations for multivariable polynomial eigenfunctions
of the quantum integrable DST model. These eigenfunctions are special
functions of the Heun class which is beyond the hypergeometric class. The
found integral equations are new and they shall provide a basis for
efficient analytical and numerical studies of such complicated functions.
From: V Kuznetsov
math.AG/9908045
Title: Selberg integral and multiple zeta values
Authors: Terasoma, Tomohide
Categories: AG Algebraic Geometry (QA Quantum Algebra)
Comments: Latex with amsart
Abstract: In this paper, we show that the coefficient of the Taylor
expansion of Selberg integrals with respect to exponent variables are
expressed as a linear combination of multiple zeta values. We use beta-nbc
base so that the Selberg integral is holomorphic with respect to the
exponent variables.
From: Tomohide Terasoma
math.QA/9907009
Title: Q-differential operators
Author: Hans Plesner Jakobsen
Categories: QA Quantum Algebra (RA Rings and Algebras)
Abstract: We set up a framework for discussing `$q$-analogues' of the
usual covariant differential operators for hermitian symmetric spaces.
This turns out to be directly related to the deformation quantization
associated to quadratic algebras satisfying certain conditions introduced
by Procesi and De Concini.
From: Hans Plesner Jakobsen
solv-int/9907001
From: matpjf@ms.unimelb.edu.au (Peter Forrester)
Date: Mon, 28 Jun 1999 06:32:30 GMT (16kb)
Classical skew orthogonal polynomials and random matrices
Authors: M. Adler, P.J. Forrester, T. Nagao, P. van Moerbeke
Comments: 21 pages, no figures
Skew orthogonal polynomials arise in the calculation of the $n$-point
distribution function for the eigenvalues of ensembles of random matrices
with orthogonal or symplectic symmetry. In particular, the distribution
functions are completely determined by a certain sum involving the skew
orthogonal polynomials. In the cases that the eigenvalue probability
density function involves a classical weight function, explicit formulas
for the skew orthogonal polynomials are given in terms of related
orthogonal polynomials, and the structure is used to give a closed
form expression for the sum. This theory treats all classical cases on an
equal footing, giving formulas applicable at once to the Hermite, Laguerre
and Jacobi cases.
From: matpjf@ms.unimelb.edu.au (Peter Forrester)
Topic #18 ------------ OP-SF NET 6.5 ----------- September 15, 1999
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