SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #3  -------------   OP-SF NET 8.5  ------------  September 15, 2001
From: OP-SF NET Editor (
Subject: Reports on OPSFA, Rome, 2001

The Sixth International Symposium on Orthogonal Polynomials, Special Functions
and Applications (OPSFA) took place at Ostia, near Roma, Italy from 18 to 22
June, 2001. Here are reports on the symposium from Tom Koornwinder, Bill Connett
and Peter Clarkson.

Some impressions by Tom Koornwinder (

This recent conference was the sixth (or by a different counting the ninth)
in a series of European meetings which started in Bar-Le-Duc, France, 1984.
The regular participants of these conferences are like relatives from a
large family, spread over Europe (or even the world), which come together
every two or three years for a joyful reunion. Serious family matters
certainly have to be discussed, but enough time should remain available for
lighter talk, for good eating and drinking and for having a lot of fun. The
cousins from Southern Europe, who are also most numerous, have in
particular excelled during this long period in being hosts to their family.

The site of the 2001 meeting was Rome, or rather Lido di Ostia, which is Rome
on the sea. Ostia is certainly less exciting than Rome (a good thing for
keeping participants at the lectures), but Rome is still close enough (a 30
minute train ride for only 1500 Lire) to make evening visits to the city by
participants or daytime visits by accompanying persons a good option. Everything,
lectures, meals and accommodation, took place in an excellent and pleasant hotel
in the middle of a large neighbourhood packed with modern apartment buildings of
moderate height. Town planners have given this neighbourhood a human aspect by
spreading shops (including many bars) all over the neighbourhood instead of
concentrating them in a shopping mall. The seaside was within 10 minutes walking

Those who were in, or passed through, Rome on the Sunday afternoon and
evening before the meeting, could give testimony of some one million people
in the streets celebrating the Italian championship of their local soccer
team Roma, after a decisive match against Parma held in the city that afternoon.
No hooligans here, no riots, no plundering, but young men with their girl
friends, and fathers and mothers with their children all happy together
about the success of their favourite club for which they had to wait so
many years. (Later I read in a Dutch newspaper that there were still some
disturbances and casualties.)

The conference had two plenary lectures every morning. Afterwards, at least
on a generic day, there were 7 contributed lectures in four parallel
sessions. The plenary lectures lasted 60 minutes including discussion, the
contributed lectures 30 minutes including discussion and possible change of
room. The plenary lectures were the following:

- A. Laforgia, M. Muldoon and P.D. Siafarikas, Commemoration of A. Elbert
- R. Askey, Solutions of some q-difference equations
- C. Dunkl, Special functions and generating functions associated with
    reflection groups
- D. Sattinger, Multipeakons and the classical moment problem
- D. Stanton, Orthogonal polynomials and identities of Rogers-Ramanujan type
- S.K. Suslov, On Askey's conjecture
- N.M. Temme, Large parameter cases of the Gauss hypergeometric function,
    in particular in connection with orthogonal polynomials
- W. Van Assche, Multiple orthogonal polynomials

Speakers in all these lectures gave excellent presentations. In the commemoration
of A. Elbert the three speakers gave a very worthy and impressive account of this
Hungarian mathematician as a person and as a scientist. As one of the speakers
said and made evident, his work was underestimated by the mathematical community.
I am regretting now that I have never been in personal contact with this
interesting mathematician, who died much too young.

While all plenary lectures were very interesting for me, I was in
particular impressed by the lectures by Sattinger and by Van Assche. David
Sattinger, coming from nonlinear pde's and integrable systems, talked
about a surprising application of the classical Stieltjes moment problem
and the related continued fraction expansion to peakon and antipeakon
solutions of the Camassa-Holm equation. The Camassa-Holm equation is a
nonlinear pde refining the KdV equation, more suitable for modelling fluid
flows in thin domains. It supports solutions, so-called peakons, that are
continuous but only piecewise analytic. Solutions with the peak downwards
are called antipeakons. During a peakon-antipeakon collision the slope
becomes infinite. Closed form of peakon-antipeakon solutions, asymptotic
behaviour and scattering shift can be obtained from the continued fraction
expansion and the corresponding orthogonal polynomials. A good reference
is R. Beals, D.H. Sattinger and J. Szmigielski, Multipeakons and the
classical moment problem, Advances in Math. 154 (2000), 229-257.

Walter Van Assche talked about multiple orthogonal polynomials. This
notion has its roots in the nineteenth century, from simultaneous rational
approximation, in particular Hermite-Pade approximation. The theory of
multiple orthogonal polynomials came up in the Eastern European literature
during the past ten or twenty years. Recently it has got a further impulse
by work of Van Assche and his collaborators. These polynomials occur in
two variants, type I and type II. Type II means for instance that a
polynomial P(x) of degree n_1+n_2+...+n_r is orthogonal to all polynomials
of degree less than n_j with respect to an weight function w_j(x) on an
interval Delta_j (j=1,...,r). One can define multiple analogues of the
classical orthogonal polynomials. In a remarkable result of Van Assche,
Geronimo and Kuijlaars the Fokas-Its-Kitaev Riemann-Hilbert problem
associated with a system of orthogonal polynomials has a generalization to
the multiple case. See "Riemann-Hilbert problems for multiple orthogonal
polynomials", to appear in "Special functions 2000: Current perspectives
and future directions", Kluwer, 2001; also downloadable from W. Van
Assche's homepage.

The contributed talks were of great variety, such that something could
be found to everybody's taste. Two contributed lectures struck me as
having deserved more emphasis by the organizers and a larger time slot. In
a brilliant 25-minute lecture Peter Clarkson gave a survey of properties
of the Painleve equations, restricting to Painleve II for the sake of
exposition. The Painleve equations may be seen as nonlinear analogues of
the classical special functions. Peter Clarkson is writing the chapter on
Painleve equations in the forthcoming NIST Digital Library of Mathematical
Functions (successor to the Handbook of Mathematical Functions by
Abramowitz and Stegun). Dan Lozier, managing editor of this DLMF project,
gave a very informative contributed lecture about the present status of
this large-scale enterprise, which will be of enormous importance for the
future of special functions usage.

A very remarkable social event was Music for Friends on Tuesday evening,
where Gino Palumbo of Universita Roma Tre, one of the conference
organizers, played piano works, partly joint with Enrico Tronci, composed
by himself during the years 1977-1987.

The organizers did a great job. Still a few critical remarks may be in
order. - The opening session was scheduled to last for one hour, but it
was finished after 15 minutes. I would have enjoyed to hear more from the
mouths of local rectors, deans and chairmen about the history of the three
Roman universities, about the number of mathematics students, about the
reason why most mathematics students in Italy are female, but most
mathematics professors are male, and whether the Museo della Matematica
housed in the Dipartimento di Matematica of Universita di Roma "la Sapienza",
and mentioned in the very comprehensive booklet Tesori di Roma, is meant as
something serious or as a kind of joke. - The topics of lectures (more so
for the contributed than for the invited lectures) remained somewhat
classical and traditional, with emphasis on one-variable theory and
analytic methods. Some more follow-up of fascinating developments about
which one could hear last year in Tempe, Arizona and some more spin-off of
things going on during last half year at the Newton Institute in
Cambridge, UK about symmetric functions and Macdonald polynomials might
have been appropriate. - A generic criticism of common practice in math
meetings is that transparencies are displayed too briefly, so that it is
impossible to take notes and to digest their full contents. During the plenary
lectures this effect might have been softened by bringing in a second projector.
At some meetings xerox copies of transparencies of plenary lectures are
distributed. A cheaper alternative might be to scan the transparencies and put
them on the web or hotel TV system. Next porno flashes on the TV's in the hotel
rooms might be replaced by flashes from the transparencies of the plenary
lectures. After paying 20000 Lire one might then get the full view of the lecture
contents on one's TV screen. - One more thing about the web. It would have been
nice if the full schedule would have been on the web some days before the
beginning of the meeting.

From: William Connett (

     Like a visitor to ancient Herculaneum in 79 A.D., who had come to
town for the very good theater, your reporter arrived in Rome on the eve
of the climactic game of the Italian Soccer cup completely unaware of the
real drama that was about to unfold, thinking only of polynomials, and the
opportunity to visit a few historical monuments (where is the Forum
anyway?) and to pay the proper obeisance to a number of Christian
monuments, St. John Lateran, St. Lawrence outside the walls, the Basilica
of Sts. Cosmos and Damian, etc., and perhaps to recite "Ode to Melancholy"
on the grave of "a Young English Poet", when like the eruption of
Vesuvius, the triumph (by the score 3-1) of AS Roma over Parma late that
Sunday afternoon (June 17), changed my world, and the pandemonium let
loose on the streets of Rome by the hundreds of thousands of hysterical
fans was the most memorable single event of the trip. I was staying in
Trastevere, and I will never forget the torrent of modern charioteers
cascading down the Lungotevere di Anguillara, singing, shouting, blowing
trumpets, waving enormous yellow and red flags from what were actually
very small motorbikes, crossing the Ponte Palatino, and pooling in the
center of Rome.

     Little did I realize, that the pool of people meant that the trolley
lines could not run, and soon the bus lines could not run either, and my
euphoria changed to the grim realization that the only way to get to the
Porta San Paola, and the train to Lido di Ostia, was to tramp, carrying my
suitcase, some three miles from Monteverde to Piramide. I caught the last
train, and collapsed amid a throng of very tired and somewhat inebriated
supporters for the thirty minute ride out to the Hotel Satellite in Lido
di Ostia.

     This was quite a dramatic beginning to the Sixth International
Symposium on Orthogonal Polynomials, Special Functions and Applications
(OPSFA-VI). In spite of a long evening of rumpus and ruction in the
streets, the some one hundred and fifty mathematicians appeared Monday
morning, for the opening ceremony of a very interesting meeting. There
were seven plenary lectures, ninety two more technical research seminars,
and an open problem session on the last day of the meeting. The following
is a very impressionistic overview of some of the highlights of the
scientific meeting. One of the topics in Dick Askey's talk which opened
the meeting, was the problem of finding bounds for the maximum values of
the polynomials in an orthogonal family. He reminded us of the argument
due to Sonine, I believe, for the Jacobi polynomials, that gives a bound
for the maximum value of each polynomial in the interval of support, and
suggested several ways that this argument might be generalized to other
orthogonal families. I was intrigued.

     The scientific committee (de Bruin, Laforgia, Marcellan, Muldoon,
Ricci, and Siafarikas) are to be congratulated for their efforts to bring
speakers to these meetings who have found new and interesting uses for the
classical mathematics. The excellent talk of David Sattinger was a good
example of this. I am still not sure what a "multipeakon" might be, but
found his application of the classical moment problem to this problem in
fluid dynamics a delight. In another direction, I was also intrigued by
the improbable idea presented by Walter Van Assche, of considering a
family of polynomials to be orthogonal with respect to two different
measures. These ideas were later elaborated on by Els Coussement, and
Jonathen Coussement in research seminars.

     I must give the prize for innovation to Franz Peherstorfer for his
very exciting talk on the distribution of the zeroes of polynomials that
are orthogonal with respect to a weight supported on disjoint intervals of
the real line. I have used a simple version of this problem for years as a
summer project for college students, and although they (and I) have
learned much from the experience, nothing I knew prepared me for the
complexity of the machinery from complex variables that he employed to
give a definitive resolution to this problem. Well done!

   Nonlinear special functions are alive and well. The talk of Peter
Clarkson did an excellent job pulling together a number of facts about the
solutions to the six Painleve equations, including the nonlinear
recurrence relations (Baecklund transformations) for the solutions of
several of them. I think that there is much more to come here. I also
enjoyed the seminar of Mohammed Sifi who did some very nice harmonic
analysis showing that the action of a particular Dunkl operator could be
considered as a multiplier that satisfied the Hormander condition, and
therefore was bounded in L^p for a range of p. Multiplier operators first
got me interested in special functions, and it is nice to see new
approaches to these old questions.

     The physical arrangements for the meeting were excellent. The hotel
was comfortable, the food quite good, and the location away from the
bustle of the center of Rome, but on the Tyrrhenian Sea, was perfect for a
scientific meeting.  In the gathering twilight, groups of mathematicians
could be seen strolling along the beach, sampling the excellent gelato
and, as is there wont, filling napkins with illegible calculations.

     Arrivederci, Roma!

From: Peter Clarkson (

The sixth International Symposium "Orthogonal Polynomials, Special
Functions and Applications" (OPSFA) was held in Rome, Italy, in June 201.
This was attended by about 150 scientists from around the world. The
plenary lectures were given by R. Askey, C.F. Dunkl, D. Sattinger, D.
Stanton, S.K. Suslov, N.M. Temme and W. van Assche. Further there was a
commemoration of A. Elbert by A. Laforgia, M. Muldoon and P.D. Siafarikas.
The contributed talks were given in four parallel sessions. The structure
of the meeting was similar to many others that I have attended. It was
extremely well organized. Locating the OPSFA meeting near to a city such
as Rome is certainly an additional attraction.

On a personal note that was the first OPSFA meeting which I have attended
and I enjoyed it very much (despite three hours delays for both my
London-Rome and Rome-London flights and missing luggage!). I had met a
number of the other participants previously at other meetings, in
particular the "Symmetries and Integrability of Difference Equations"
series of meetings (Esterel, Canada, 1994; Canterbury, UK, 1996; Sabaudia,
Italy, 1998; Tokyo, Japan 2000). My own research field is the study of
nonlinear differential equations and nonlinear difference equations, in
particular exact solutions and asymptotics. Frequently we use results from
"Special Functions" and "Orthogonal Polynomials", despite being linear
equations in the solution of the nonlinear problems. I was pleased and
encouraged that there were some talks on nonlinear problems at this
meeting, including a plenary lecture by David Sattinger.

I believe that there are many mathematicians and physicists who have
research interest in "Special Functions" and "Orthogonal Polynomials"
though not as their main field of research. I feel that the involvement of
such scientists in the OP-SF activity group and participation in future
OPSFA meetings should be strongly encouraged.

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