o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o July 15, 2000 O P - S F N E T Volume 7, Number 4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Editor: Martin Muldoon muldoon@yorku.ca The Electronic News Net of the SIAM Activity Group on Orthogonal Polynomials and Special Functions Please send contributions to: poly@siam.org Subscribe by mailing to: poly-request@siam.org or to: listproc@nist.gov o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o Today's Topics: 1. From the Editor 2. Conference on q-series 3. Second Announcement of the SIDE IV Meeting 4. Workshop on Quasiclassical and Quantum Structures 5. 2001: A Mathematics Odyssey 6. Dortmund meeting on Approximation Theory 7. Reports on Special Functions 2000 8. Future Directions in Special Functions 9. 8th International Krawtchouk Conference 10. Jose J. Guadalupe (1946-2000) 11. Authors Selected for NIST Digital Library Project 12. Special Functions Posters 13. OP-SF preprints in xxx archive 14. Changes of address, WWW pages, etc. 15. About the Activity Group 16. Submitting contributions to OP-SF NET and Newsletter Calendar of Events: 2000 July 19-26: Third World Congress of Nonlinear Analysts, Catania, Italy (including session on "Adaptive quadrature and cubature formulae". 7.1 #6 July 24-28: Summer School "Orthogonal Polynomials and Special Functions", Laredo, Spain. 6.6 #3 Dedicated to Jose Javier Guadalupe August 5-8: International Symposium on Analysis, Combinatorics and Computing, Dalian, China 7.1 #7 August 14-18: International Symposium on Applied Mathematics, Dalian, China 6.5 #5 September 22-28: International Conference on Functional Analysis and Approximation Theory, Acquafredda di Maratea, Italy 7.2 #6 October 26-28: q-Series with Applications to Combinatorics, Number Theory and Physics, University of Illinois, USA 7.4 #2 November 27 - December 1: 4th International Interdisciplinary meeting on "Symmetries and Integrability of Difference Equations", Tokyo, Japan. 7.4 #3 2001 January 9-14: Workshop on Quasiclassical and Quantum Structures, Fields Institute, Toronto, Canada 7.4 #4 June 18-22: Symposium on Orthogonal Polynomials, Special Functions and Applications, Rome, Italy 7.3 #2 August 6-10: Analytic theory of continued fractions, orthogonal functions and related topics, Grand Junction, Colorado, USA 7.4 #5 August 20-24: 3rd International meeting on Approximation Theory, Dortmund, Germany 7.4 #6 October 1-5: "Numerical Algorithms", Conference in Honor of Claude Brezinski, Marrakesh, Morocco 7.3 #3 Future plans: As already mentioned in OP-SF NET 6.5, the next meeting in the series Fields-Toronto (1995) - CRM-Montreal (1996) - Mount Holyoke (1998) - Hong Kong (1999) - Arizona (2000) is expected to be held in Amsterdam, in 2002, probably in early summer, to be organized by Tom Koornwinder (thk@uwa.wins.nl), Nico Temme (nico@cwi.nl) and Erik Koelink (koelink@twi.tudelft.nl). Topic #1 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: From the Editor Since the appearance of our last issue the major NATO ASI and Conference has taken place in Tempe, Arizona. In this issue, we feature reports from some of those attending (Erik Koelink, Kathy Driver and Bill Connett) (Topic #7) as well as a detailed report on the session on "Future Directions" by Walter Van Assche (Topic #8). After some discussion between the Officers of the Activity Group, I decided to reduce the information in "OP-SF preprints in xxx archive" to simply giving abstract numbers, authors, titles and e-mail addresses. We continue to discuss how much detail to include in conference announcements. It has been suggested that it would be sufficient to include titles, location and dates with a link to the conference web page. On the other hand, there seems to be a number of readers for whom web access is not yet fast and efficient so that there remains a need for more information on conferences in OP-SF NET. I continue to ask for a volunteer or volunteers to take over as Editor from January 1, 2001. Topic #2 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: Conference on q-series The following is a selection of the information on the web site: http://www.math.wisc.edu/~ono/qseries.html q-series with Applications to Combinatorics, Number Theory and Physics. October 26-28, 2000 University of Illinois at Urbana-Champaign. Confirmed Plenary Speakers Scott Ahlgren (Colgate University) George Andrews (Penn State University) Richard Askey (University of Wisconsin) Anne Schilling (MIT) Dennis Stanton (University of Minnesota) Confirmed Invited Speakers Krishnaswami Alladi (University of Florida) Douglas Bowman (University of Illinois) Thomas Ernst (Uppsala University) Mourad Ismail (University of South Florida) Christian Krattenthaler (University of Vienna) Jeremy Lovejoy (University of Wisconsin) John McKay (Concordia University) Steve Milne (Ohio State University) Katsuhisa Mimachi (Kyushu University) Morris Newman (University of California, Santa Barbara) Peter Paule (University of Linz)-tentative Sasha Polishchuk (Boston University) Mizan Rahman (Carleton University) Ole Warnaar (University of Amsterdam) - tentative Sander Zwegers (University of Utrecht) Registration: Registration information will be available soon. To be placed an an e-mail list, send an e-mail to berndt@math.uiuc.edu Financial support is available to a limited number of participants with some preference given to graduate students and new PhD's. To apply for this support, send e-mail to ono@math.wisc.edu by September 1, 2000. Scientific Organizer Bruce Berndt and Ken Ono Topic #3 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Tetsuji Tokihiro Subject: Second Announcement of the SIDE IV Meeting 4th International Interdisciplinary Meeting on "Symmetries and Integrability of Difference Equations" Tokyo (Japan), 27 November - 1 December 2000 The SIDE meetings are intended to provide a point of contact between researchers of various disciplines, all working or using methods from discrete systems, i.e. systems that can be described by ordinary or partial difference equations. This domain forms the core of a great variety of fields, including classical and quantum physics, computer science, mathematical biology, economics, numerical analysis, discrete geometry, and so on. The main topics of the present meeting will be: Integrable difference equations, symmetries of ordinary and partial difference equations, cellular automata, discrete monodromy problems, q-special functions, discrete geometry, applications to physics and engineering. In this meeting, lectures will be delivered in the auditorium of the Graduate School of Mathematical Sciences, University of Tokyo. (Information is available on http://liaison.ms.u-tokyo.ac.jp/) Since our idea is to keep to a single session format, we plan to accept only a restricted number of applications. All of the talks will be from 20 to 30 minutes long. We will also organize poster sessions. The cost of participation consists of a registration fee (including excursion and banquet) of 15,000 Japanese-yen. As for the accommodation, we are happy to provide reservations in the hotel: HILPORT HOTEL Sakuraoka-cho 23-19, Shibuya-ku, Tokyo 150-0031, Japan Tel. (+81)3-3462-5171 Fax. (+81)3-3496-2066 at the price of 12,000 Japanese yen (all inclusive, i.e. breakfast, lunch and dinner) or 9,500 Japanese yen (with breakfast only) per day. The following list is the expected participants at present: Mark ABLOWITZ (Colorado University, USA) Vsevolod ADLER (Ufa Institute of Mathematics, Russia) Claude BREZINSKI (Universite' des Sciences et Technologies de Lille, France) Robert CONTE (CEA--Saclay, France) Adam DOLIWA (Warsaw University, Poland) Claire GILSON (University of Glasgow, UK) Basile GRAMMATICOS (Universite' Paris VII, France) Valeri GROMAK (Belarus State University, Belarus) Jarmo HIETARINTA (University of Turku, Finland) Ryogo HIROTA (Waseda University, Japan) Xing-Biao HU (Academia Sinica, China) Mourad ISMAIL (University of South Florida, USA) Michio JIMBO (University of Tokyo, Japan) Nalini JOSHI (University of Adelaide, Australia) Kenji KAJIWARA (Doshisha University, Japan) Rinat KASHAEV (Steklov Math. Institute, Russia) Boris KONOPELCHENKO (Universita di Lecce, Italy) Martin KRUSKAL (University of Rutgers, USA) Franklin LAMBERT (Vrije Universiteit Brussel, Belgium) Decio LEVI (Universita' di Roma Tre, Italy) Sergey LEBLE (Technical University of Gda\'nsk, Poland ) Yoshimasa NAKAMURA (Osaka University, Japan) Atsushi NAGAI (Osaka University, Japan) Frank NIJHOFF (University of Leeds, UK) Jon NIMMO (University of Glasgow, UK) Katsuhiro NISHINARI (Ryukoku University, Japan) Masatoshi NOUMI (Kobe University, Japan) Yasuhiro OHTA (Hiroshima University, Japan) Kazuo OKAMOTO (University of Tokyo, Japan) Reinout QUISPEL (Latrobe University, Australia) Orlando RAGNISCO (University of Roma, Italy) Alfred RAMANI (Ecole Polytechnique, France) Jean-Pierre RAMIS (Universite' Toulouse, France) Simon RUIJSENAARS (CRM, Netherlands) Hidetaka SAKAI (University of Tokyo, Japan) Paolo SANTINI (University of Roma, Italy) Wolfgang SCHIEF (University of New South Wales, Australia) Serguei SERGUEEV (BLTP JINR, Russia) Evgueni SKLYANIN (Steklov Institute of Mathematics at St. Petersburg, Russia) Juri SURIS (Technische Universitaet Berlin, Germany) Daisuke TAKAHASHI (Waseda University, Japan) Munirathinam TAMIZHMANI (Pondicherry University, India) Morikazu TODA (Honorary Chairperson) Walter VAN ASSCHE (Katholieke Universiteit Leuven, Belgium) Ralph WILLOX (University of Tokyo, Japan & Vrije Universiteit Brussel, Belgium) Pavel WINTERNITZ (Universite de Montreal, Canada) Youjin ZHANG (Tsinghua University, China) If you are interested in attending, please visit the web site: http://elrond.doshisha.ac.jp/side4/index.html where you can find the application form which should be sent to us. Information updates will be available on this web site. Alternatively, you can get an application form from the organizers. Postal address: SIDE IV Graduate School of Mathematical Sciences University of Tokyo 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, JAPAN Fax: (+81) 3-5465-8312 email: side4-org@elrond.doshisha.ac.jp website: http://elrond.doshisha.ac.jp/side4/index.html email: side4-org@elrond.doshisha.ac.jp Local organizers are: J. Satsuma, T. Tokihiro (Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo153-8914, Japan) e-mail: satsuma@poisson.ms.u-tokyo.ac.jp toki@poisson.ms.u-tokyo.ac.jp fax: +81-3-5465-8312 Topic #4 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: Workshop on Quasiclassical and Quantum Structures >From http://www.fields.utoronto.ca/lt-qq.html Workshop on Quasiclassical and Quantum Structures Tuesday, January 9 - Sunday, January 14, 2001 at the Fields Institute Toronto, Ontario, Canada Organizers: Pavel Etingof, Massachusetts Institute of Technology Boris Khesin, University of Toronto Topics include: - Classical and quantum integrable systems - Macdonald theory - Poisson-Lie groups, quantum groups, dynamical quantum groups, and quantization - Infinite-dimensional Lie algebras and structures, and their quantum deformations - q-Virasoro, q-W-algebras and their quasiclassical limits, affine and quantum affine algebras at the critical level - Quantization of Poisson manifolds - Hypergeometric and q-hypergeometric functions, their generalizations, KZ, qKZ, KZB, qKZB equations, Elliptic quantum groups Limited funds may be available to assist graduate students and postdoctoral participants. Please contact the organizers by fax at: (416) 348-9759, or through e-mail at: lt-structure@fields.utoronto.ca. All are welcome. This Workshop is part of the "Infinite-dimensional Lie Theory and its Applications" and "Symplectic Geometry, Topology, and Gauge Theory" programs, both hosted by the Fields Institute in Fall 2000 and Spring 2001, respectively. Contact mailing address: c/o Lie Theory, The Fields Institute 222 College Street, Toronto, Ontario M5T 3J1 Telephone: (416) 348-9710 Fax: (416) 348-9759 Topic #5 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Phil Gustafson Subject: 2001: A Mathematics Odyssey FIRST ANNOUNCEMENT 2001: A Mathematics Odyssey a conference on the analytic theory of continued fractions, orthogonal functions, rational approximation and related topics. A Celebration of the 70th birthday of Dr. William B. Jones Professor Emeritus, University of Colorado, Boulder, USA In recognition of the contributions Professor William B. Jones has made to the field of continued fractions and rational approximation, we are pleased to announce a conference organized in his honor. The conference will be held August 6-10, 2001, at Mesa State College in Grand Junction, Colorado, USA. We invite contributions from both the theoretical and computational aspects of continued fractions, orthogonal polynomials, rational approximation, and related areas and applications. There is no need to commit to attending the conference at this time. However, if you are interested in receiving a second announcement and would like to be on our mailing list, please respond or email to one of the organizers at the address below, including your name, mailing address and email address. More information about Mesa State College and Grand Junction, Colorado, can be found at http://www.mesastate.edu, and http://www2.mesastate.edu/community_links.htm. We hope to see you there. Organizers: Cathy Bonan-Hamada, Phil Gustafson Mathematics Department Mesa State College 1100 North Ave. Grand Junction, CO 81501-3122 USA cbonan@mesastate.edu, pgustafs@mesastate.edu Topic #6 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: IDoMAT 2001 - Dortmund meeting on Approximation Theory From: http://www.mathematik.uni-dortmund.de/lsviii/idomat2001.html 3rd INTERNATIONAL DORTMUND MEETING APPROXIMATION THEORY IDoMAT 2001 August 20 - 24, 2001 Haus Bommerholz - Witten, Germany. Organizers: Martin D. Buhmann, University of Giessen martin.buhmann@math.uni-giessen.de Detlef H. Mache, University of Dortmund, mache@math.uni-dortmund.de Manfred W. Müller, University of Dortmund. mueller@math.uni-dortmund.de The main aim of this conference IDoMAT 2001 is to bring together invited researcher, to discuss problems and to promote the transfer of results, ideas and applicable methods in the following fields in the Theory of Constructive Approximation: Approximation Methods, Approximation by Operators, Interpolation Radial Basis Functions Orthogonal Polynomials (Multi-) Wavelets, Neuronal Networks, CAGD Proceedings of IDoMAT 2001 and accepted research papers: We intend to publish the invited lectures and the accepted research papers in the Proceedings (Volume 3): New Topics in Constructive Approximation. This third Volume (after Volume 1: Approximation Theory - IDoMAT 95 (Akademie Verlag Berlin) and Volume 2: New Developments in Approximation Theory - IDoMAT 98 (Birkhäuser Verlag Basel)) will be published in the International Series of Numerical Mathematics by Birkhäuser Verlag Basel. IDoMAT 2001 - Office: University of Dortmund Institute of Applied Mathematics (Approximation Theory, LS VIII) D - 44221 Dortmund (Germany) E-mail: idomat@math.uni-dortmund.de Topic #7 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: Reports on Special Functions 2000 Special Functions 2000: Current Perspective and Future Directions, Arizona State University, May 29 to June 9, 2000. 1. From Erik Koelink The 2-week conference actually consisted of three parts. A NATO Advanced Study Institute, a NSF Research Conference and a series of lectures on Computer Algebra. The NATO ASI talks were plenary 1-hour talks on various subjects. These talks ranged from introductory talks to more advanced talks on recent results. To mention a few, I very much liked the talks by Christian Krattenthaler (a great performer with transparencies and figures), Mizan Rahman (associated orthogonal polynomials and Askey-Wilson operators), Simon Ruijsenaars (solutions of the Askey-Wilson difference operators with q on the unit circle), Percy Deift (Riemann-Hilbert problems, and their application to all kinds of problems), Ken Ono (recent exiting results in number theory), Slava Spiridonov (a very impressive account of factorisations and their use), Sergei Suslov (q-Fourier series and a q-Riemann zeta function), Alexei Zhedanov (biorthogonal rational functions), Hjalmar Rosengren (dynamical Yang-Baxter equation and n-j symbols, n=3,6,9). The above list consists more of the talks on more recent results, and there were also some very good introductions by Dennis Stanton and Mourad Ismail. Maybe I should mention all speakers in this programme, since the talks were in general very good and very interesting. The half-hour talks in the NSF-programme were organised in parallel sessions, so that it's impossible to attend them all. Some of my personal favourites were Jan Felipe van Diejen on a multivariable summation formula for elliptic hypergeometric series conjectured by Ole Warnaar that he could almost prove, Katsuhisa Mimachi on representations of the Hecke algebra on twisted homology, Andre Unterberger on relativistic quantisation applied to special functions, Michitomo Nishizawa on all kinds of generalisations of the gamma function and Joaquin Bustoz on q-Bessel functions and q-Lommel polynomials. The talks in the Computer Algebra part were usually scheduled in the evening, which is one of the reasons that I missed a number of them. Some of these talks were presentations by people from Mathematica who discussed their huge posters on special functions. The chief of the (local) organisation was Sergei Suslov and he has made a tremendous effort in making the conference such a success. His daughter Liliya has been a great help in organising. All in all the organising committee has done a very nice job. The Tempe surroundings were very pleasant, but also very hot. The Grand Canyon was one of the touristic events and, being an inhabitant of a flat country, I was really impressed with it. 2. From Kathy Driver <036KAD@cosmos.wits.ac.za> Over 100 mathematicians gathered in Tempe to discuss Current Perspectives and Future Directions in the area of Special Functions. The meeting was remarkable from several different perspectives, perhaps the most striking feature being the diversity of areas in which talks were presented. The old maxim that "special functions are everywhere" gained considerable credibility as a variety of topics unfolded both in the main presentations and also during the parallel sessions. Orthogonal polynomials , special functions of one and several variables, asymptotics, continued fractions, applications to number theory, combinatorics and mathematical physics, integrable systems, harmonic analysis and quantum groups, Painleve classification were listed as some of the topics to be covered and that was no exaggeration--these and many others featured in a lively and well- organised programme. Many of the well-established masters in the area presented talks, mostly for two separate hours which facilitated more than just a glimpse of their ideas and expertise and attendance by graduate students was noteworthy. Richard Askey commented in his speech at the banquet that he was grateful to those present for carrying the banner of special functions forward over the past ten years and it was easy to see why he is pleased with developments in the field. The venue was comfortable and suitable, the organisational details were taken care of in exemplary fashion and the power failure was thankfully short in duration, given the formidable heat in the desert at that time of year. attended. This was an extremely successful meeting and bodes very well for continuing vigorous interest in this area. 3. From William Connett It was hot. The sun was a terrifying presence. Your reporter would look out the window of the conference center and he could see for two miles down Apache Boulevard, and often not a single person could be seen in the open in this very modern city. Although the temperature of the air was over 100 F, nobody went into the swimming pool during the day because the temperature of the pool was above 90 F and the sun was so intense that it would give you skin cancer in five minutes. On the other hand it was not the hottest mathematical meeting that your reporter ever attended. I remember one epic meeting in Morocco in July when there was no air conditioning in the hotel, no water in the rest rooms, the temperature one day got up to 130F, and all the lectures were in French. By that standard, this meeting was a cake walk. The Holiday Inn was a very pleasant venue. There was so much air conditioning in the conference hall that most participants wore their jackets, the food was serviceable and easy to obtain, and the lay out of the conference with all lectures, food, and rooms in one location made it a very pleasant meeting. The weather kept all the participants in the motel, so that mathematical conversations were spontaneous and quite easy. This was one of the most complex meetings that I have ever attended. It was concurrently: first, a NATO funded Advanced Study Institute, second, a NSF funded Research Conference, and third, a mini conference on Computer Algebra and Special Functions on the Web, supported by Wolfram Research and other sources. This may become the new paradigm for organizing a conference. The field of special functions has grown so enormously that it is difficult to remember the time when the few enthusiasts could easily fit into a small seminar room to discuss the problems of common interest. Now there is a cast of hundreds, working in dozens of areas. The specialty meeting now take on more of the character of the large national meetings. And the total experience was quite enjoyable. The NATO funded Advanced Study Institute featured a number of hour long talks which were intended to introduce a topic, and bring a sophisticated audience up to a certain level of competence on a particular problem. For example, Luc Vinet gave two lectures entitled "Advances in multivariable special functions and mathematical physics", but actually he had the courage to ignore the physics, and work through several concrete examples of the new families of symmetric polynomials called atoms, related to the t-Kostka polynomials. The examples were carefully done, and the audience was very appreciative of the care with which they were explained. Two other speakers in this section that I really enjoyed were Christian Krattenthaler who gave a lovely series of lectures on plane partitions, orthogonal polynomials, and hypergeometric series. Christian certainly wins the prize for the most innovative use of the overhead projector in his presentations. Even if I did not enjoy the topic, I would be fascinated by his implementation of ur-animation in his talks. His screen reminds me of some of the early Loony Toons cartoons with the jerky but eye-catching animation. The other speaker was Alexander Kitaev, who introduced the audience to the six versions of the Painleve equation and their solutions. I was very appreciative of his effort to explain to the outsider what was going on in this important area. The NSF research Conference included many more traditional research type talks, from this feast of topics, I will mention two that I found particularly memorable: Yuan Xu talked about problems in Fourier expansions in several variables, and Khalifa Trimeche worked out the harmonic analysis associated with a singular differential-difference operator (a generalization of the Dunkl operator on the real line). There were many other excellent talks. The final part of the conference were the sessions on computer algebra. real indication of the interest in these topics (or perhaps just the weather) that even though the meetings started at 8:00am and went all day with only an hour for lunch and dinner, there would frequently be over one hundred people in the lecture hall at 9:00pm to hear Oleg Marichev or Michael Trott from Mathematica talk about their product, or Lance Littlejohn or Axel Riese talk about some new software that they had produced to simplify certain calculations. The wealth of computational tools now available is truly impressive. Many talks were given in many areas, and this brief note can only mention a few of them. On the other hand, I think it is important to try and see what the new tools or new areas where great progress is being made. I will mention three. First, it is quite clear from the talks of Dunkl, Xu, Littlejohn, Kill, Haine and others that finally a theory of multivariable polynomials is beginning to emerge. We may not agree on which of these polynomials to call classical, but we are beginning to see the clear lines of the theory. I look forward to the new book from Dunkl and Xu. Second, it was clear from the talks of Percy Deift and Walter Van Assche that the techniques developed to solve the Riemann-Hilbert problem are providing powerful new tools for the study of orthogonal polynomials. Finally, I have gone to many meeting and never heard mentioned the solutions of the Painleve equation. Such solutions were not on everyone's lips at this meeting, but they were mentioned in at least five different talks, and they were the subject of two hours of plenary talks. We will hear much more about "the Painleve Transcendents". Finally we must thank the organizing committee: Bustoz, Ismail, Koornwinder, Spiridonov, Suslov, and Vinet for a splendid program, and the gracious hosts from Arizona State University, Sergi Suslov and Joaquin Bustoz for a wonderful scientific adventure in a very hot corner of the world. Hot mathematics in a hot place! Topic #8 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Walter Van Assche Subject: Future Directions in Special Functions [From the June Newsletter] On the last day of the NATO Advanced Study Institute on "Special functions 2000: Present Perspectives and Future Directions" (Tempe, Arizona, May 29 - June 9, 2000) there was a session on Future Directions, chaired by Richard Askey. The following is an attempt to summarize what was said. First Askey gave some _advice_. - Ramanujan is still a very big source of future research, especially regarding congruences for the partition function. Exciting new results have been found by Ken Ono, but what has been found is probably only a hint of what else will be discovered. - Other indications that there is still a lot to be learned from a study of Ramanujan is the recent work on elliptic functions with different bases, which was probably the first use of cubic transformations of hypergeometric functions, Ramanujan's wonderful series for 1/\pi and the remarkable identities found in the lost notebook (including results on mock theta functions). - A second source of problems is in the work of David and Gregory Chudnovsky. They have mentioned many problems and results, some of which are eventually published, but many have not been published. Their papers are worth studying, although this is not easy. - Work of Rodney Baxter led to the discovery of quantum groups and was one of the sources for elliptic hypergeometric functions. There is much more there which needs to be understood. - Bill Gosper has sent e-mail containing many interesting formulas to many people. Some e-mails have been understood, but many still are full of mysteries. He then continued with some _safe_predictions_: - Special functions of several variables will be studied extensively (orthogonal polynomials, hypergeometric and basic hypergeometric functions, elliptic hypergeometric functions). - Cubic transformations will get more attention (see, e.g., Bressoud's treatment of alternating sign matrices). - There will be much more combinatorial work. - Computer algebra will become important but will not replace thinking. - Nonlinear equations and special functions (Painleve) will receive more attention. - Regarding asymptotics, there will be a deeper understanding in one variable, there will be much more on difference equations, and asymptotics for several variables will be developed more fully. Askey then expressed some _hopes_: - Special functions in infinite dimensional spaces will appear. - Linear differential equations with more than three regular singular points will be understood better than at present. - Special functions over p-adic and finite fields become more popular. - Orthogonal (and biorthogonal) rational functions will start to have more applications. - Understanding mock theta functions via mock modular functions will partly succeed. - The location of zeros of _2F_1(a,b;c;z) on (-\infty,0), (0,1), and (1,\infty) in the terminating case is known (also in the complex plane). We need extensions to _3F_2 and _2\phi_1 and other (basic) hypergeometric functions. Finally Askey mentioned some _wild_guesses_: - Cubic transformations for hypergeometric functions really live in double series associated to G_2 and we are only seeing one dimensional parts of this. - The function G satisfying the relation G(x+1) = \Gamma(x) G(x) has an integral representation, probably an infinite dimensional one (a limit of Selberg's integral?). - 9-j symbols as orthogonal polynomials in two variables can be represented as a double series. Some other participants added some other interesting observations and suggestions for future work. Tom Koornwinder: - Matrix valued special functions. An obvious source of such functions are the generalized spherical functions associated with Riemannian symmetric pairs (G,K) and higher dimensional representations of K. See Grunbaum's lecture at this meeting for the example (SU(3),SU(2)). - Orthogonal polynomials depending on non-commuting variables naturally occur in connection with quantum groups, see for instance the q-disk polynomials studied by Paul Floris, which are spherical functions for the quantum Gelfand pair (U_q(n), U_q(n-1)). More examples should be obtained and a general theory of such polynomials should be set up. - Special functions associated with affine Lie algebras. Remarkable interpretations of special functions have already been found on affine Lie algebras (see the book by Victor Kac), but much more should be possible here. The lecture by Paul Terwilliger at this meeting gives some hints in this direction. - The work of Borcherds: generalized Kac-Moody algebras, vertex algebras and lattices in relationship with automorphic functions. - Algebraic and combinatorial techniques in contrast with analytic techniques have quickly gained importance in work on (q-)special functions during the last few decades. Algebra often gives rise to quick and easy formal proofs of, for instance, limit results. Usually, a rigorous analytic proof is much longer, while it does not give new insights. In fact, the rigorous proof is often omitted. There is need for a meta-theory which explains why formally obtained results are so often correct results. Vyacheslav Spiridonov: - It is likely that important special functions are hidden in some of the work on differential-delay and differential-difference equations. - Development of elliptic special functions (elliptic beta integral, elliptic deformations of Painleve). - Connections of our work with other fields (biology, economy, etc.). - Wavelets could be studied as special functions. - Ismail's q-discriminant needs an interpretation in statistical mechanics. Stephen Milne and Tom Koornwinder: - The lecture by Jan Felipe van Diejen and the discussion after Stephen Milne's last lecture at this meeting made clear that several different types of multivariable analogues of one-variable (q-)hypergeometric series have been studied extensively, but that their mutual relationship is poorly understood. The three most important types are: 1. Explicit series associated to classical root systems (Biedenharn, Gustafson, Milne), 2. Hecke-Opdam hypergeometric functions and Macdonald polynomials associated to any root system ((q-)differential equations, usually no explicit series), 3. Gelfand hypergeometric functions (again (q-)differential equations, usually no explicit series). Van Diejen, in his lecture, added to this list: 4. hypergeometric sums of q-Selberg type, 5. hypergeometric sums coming from matrix inversion. Koornwinder would like to add: 6. Solutions of KZ(B) and q-KZ(B) equations, 7. 3-j, 6-j and 9-j symbols for higher rank groups. - Elliptic generalizations of one and multivariable hypergeometric functions are also coming up now. Stephen Milne added that it is likely that the concept "very well poised" ties these various types of multivariable functions together. - Applications in combinatorics and number theory are welcome. Sergei Suslov: - One needs to understand the classical q-functions, beginning with the q-exponential and q-trigonometric functions. - Orthogonal q-functions (also the non-terminating series) and special limiting cases are useful. - Biorthogonal rational functions are a rich source of research problems. Mourad Ismail: - There is still a lot of work to be done in moment problems and continued fractions, in particular indeterminate moment problems. - Discriminants, lowering operators and electrostatics, such as the Coulomb gas model. - Multivariate extensions. Walter Van Assche: There is still quite some work in orthogonal polynomials: - The asymptotic zero distribution and logarithmic potential theory (with external fields and constraints) has been worked out in quite some detail now. For some q-polynomials one seems to need circular symmetric weights. We don't know how to handle big q-Jacobi, big q-Laguerre, q-Hahn and q-Racah yet. - There is a well established theory for strong asymptotics of orthogonal polynomials on the unit circle and on the interval [-1,1] (Szego's theory). The analog of this theory for the infinite interval (e.g., Freud weights) is starting to become clear. So far there is no theory for orthogonal polynomials on a discrete set (such as the integers). The Riemann-Hilbert technique may be useful here. - Multivariate orthogonal polynomials need more attention. - Multiple orthogonal polynomials (one variable but several weights) may be a rich source of nice research. Some of these multiple orthogonal polynomials can be written in terms of nice special functions (generalized hypergeometric functions, hypergeometric functions of several variables, etc.). The analysis involves Riemann surfaces with several sheets, equilibrium problems for vector potentials, banded non-symmetric operators. We already know some nice applications in number theory and dynamical systems. Other applications would be nice. - Higher order recurrence relations and asymptotics for solutions of difference equations are useful. George Gasper: Positivity proofs and proofs that certain functions only have real zeros are very useful. Erik Koelink - The _8\phi_7 basic hypergeometric is very nice and the multivariate case would be even nicer. - Where do the elliptic hypergeometric functions of Frenkel and Turaev live? - Is there a way to use Riemann-Hilbert problems for quantum groups? - Applications of multivariate orthogonal polynomials in probability theory. This is just a brief description and a personal account of what was said during the session on future directions. Some other participants added some open problems, but it would take too much space to report on these in the newsletter. Topic #9 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Vadim Zelenkov Subject: 8th International Krawtchouk Conference The Eighth International Krawtchouk Conference was held in the period May 11-14, 2000. The conference was organized by the Institute of Mathematics (National Academy of Sciences of the Ukraine), the Kiev National Shevchenko University, the National Drahomanov Pedagogical University and the National Technical University of the Ukraine (KPI). It took place in Kiev (Kyiv), the capital of the Ukraine. The 626 participants represented Algeria, Armenia, Australia, Belarus, Italy, Kazakhstan, Lithuania, Russia, Ukraine and USA. Following tradition, the Conference included four sections: - Differential and integral equations, and applications - Algebra, geometry. Mathematical and numerical analysis - Theory of probability and mathematical statistics - History, methods of teaching of mathematics The titles of the reports which are most relevant to orthogonal polynomials and special functions (organized "by functions") are: Savva V.A., Khlus O.V.: Krawtchouk Quantum Oscillator: Dynamics Features Groza V.A.: The Quantum Group SU_q(2) and Product Formula for q-Krawtchouk Polynomials Zelenkov V.I.: Orthogonal Polynomials Given by Recurrence Relation Mamteev J.A., Huchraeva T.S., Burjacov A.N.: The Solution of a Contact Problem Using the Modified Struve and Bessel Functions Ivcina A.E., Huchraeva T.S., Stukalina V.I.: On Modified Struve Function and the Principal Characteristic Mamteev J.A., Stukalina V.I.: Modified Struve Function L_\nu(z) and Struve Function H_\nu(z) Markova K.V.: Inversion Formula for Hankel Transform for a Class of Functions Gaidey V.O.: On Generalization of Bessel Function Bilyk Yu.: On Multiplication Theorem for Generalized Hypergeometric Functions Warren D., Seneta E.: Hypergeometric Polynomial Probability Generating Functions Nikitina O.M.: Finite Hybrid Integral Transforms of Mehler-Fock Type of the First Kind Romanenko N.V.: Fourier Series with Mathieu Functions Timan M.F.: On Fourier Series with Monotonic Coefficients Tretyakova N.N.: Limit Relations Between Some Integral Transforms Yakubovich S.B.: On the Titchmarsh Integral Transformation The book of abstracts contains 560 pages. As in the previous conferences the opening ceremony was dedicated to the memory of M. Krawtchouk. A memorial booklet "Son of the Sky" was presented by Galina Datsyuk and Mikola Soroka. The second book published on the eve of the conference is the collection of Krawtchouk's popular scientific works. It includes in particular studies in the history of mathematics (e.g. Euler's Influence on the further Development of Mathematics), popular physical articles (Space, Time, Matter) scientific reports and lyrical notes about the author's travel to the World Mathematical Congress in Bologna and others. The book also contains the biography of M. Krawtchouk written by Prof. Nina Virchenko who has provide and inspired investigations of Krawtchouk's life and work for many years. During the exciting tour of Kiev, the conference participants stood for a minute of silence near the newly opened memorial plaque on the house where M. Krawtchouk lived and where he was arrested on February 21, 1938. The 9th Krawtchouk Conference will take place in 2002 - the 110th anniversary of the birth of M. Krawtchouk. All the necessary information will appear on the Krawtchouk Polynomials Home Page: http://www.isir.minsk.by/~zelenkov/physmath/kr_polyn Topic #10 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Manuel Alfaro Subject: Jose J. (Chicho) Guadalupe (1946 - 2000) (From the Activity Group's June Newsletter) Prof. Jose Javier Guadalupe (Chicho) died on April 1, 2000 at the age of 54 in a car accident. He was born in Santa Cruz de la Palma (Canary Islands) and studied at the University of Zaragoza, Spain. He worked at the University of Zaragoza (1970-1992) and the University of La Rioja (1992-2000). Chicho was a student of Jose Luis Rubio de Francia under whose supervision he prepared his Ph.D. on "Closure in L^p(\mu) of analytical polynomials in the unit circle" at the University of Zaragoza. His general area of research was harmonic analysis. His early work was on closure of analytical polynomials on weighted Jordan curves. Later he worked on Fourier series in orthogonal polynomials and special functions. Recently, he was interested in Stieltjes polynomials and varying measures. He was a very active man, and an organizer of mathematical activities. In the field of orthogonal polynomials and special functions, Chicho promoted the idea of having a series of Spanish Symposia Symposia of Orthogonal Polynomials and Applications. The first Symposium was organized by him in Logrono in 1983. His death is a great loss for his colleagues and mainly for Spanish people working on orthogonal polynomials and special functions. (Editor's Note: The Summer School "Orthogonal Polynomials and Special Functions" to be held in Laredo, July 24-28, 2000 (OP-SF NET 6.6, Topic #3) is to be dedicated to the memory of Chico Guadalupe.) Topic #11 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Daniel Lozier Subject: Authors Selected for NIST Digital Library Project The National Institute of Standards and Technology (NIST) has selected authors for the following chapters of the Digital Library of Mathematical Functions (DLMF): Mathematical and Physical Constants. NIST. Algebraic and Analytical Methods. R. Askey & R. Roy. Asymptotic Approximations. F. Olver & R. Wong. Numerical Methods. C. Brezinski & W. Gautschi. Computer Algebra. P. Paule & F. Chyzak. Elementary Functions. S. Krantz. Gamma Function. R. Askey. Exponential Integral, Logarithmic Integral, Sine and Cosine Integrals. N. Temme. Error Functions, Dawson's Integral, Fresnel Integrals. N. Temme. Incomplete Gamma Functions and Generalized Exponential Integral. R. Paris. Airy and Related Functions. F. Olver. Bessel Functions. F. Olver & L. Maximon. Struve Functions and Anger-Weber Functions. R. Paris. Confluent Hypergeometric Functions. J. Wimp. Coulomb Wave Functions. M. Seaton. Parabolic Cylinder Functions. N. Temme. Legendre Functions and Spherical Harmonics. M. Dunster. Hypergeometric Functions. A. Olde Daalhuis. Generalized Hypergeometric Functions and Meijer G-Function. R. Askey. q-Hypergeometric Functions. G. Andrews. Classical Orthogonal Polynomials. R. Koekoek & R. Swarttouw. Other Orthogonal Polynomials. R. Koekoek & R. Swarttouw. Elliptic Integrals. B. Carlson. Jacobian Elliptic Functions and Theta Functions. P. Walker & W. Reinhardt. Weierstrass Elliptic Functions. P. Walker & W. Reinhardt. Bernoulli and Euler Numbers and Polynomials. K. Dilcher. Zeta and Related Functions. T. Apostol. Combinatorial Analysis. D. Bressoud. Functions of Number Theory. T. Apostol. Statistical Methods and Distributions. I. Olkin & D. Kemp. Mathieu Functions and Hill's Equation. G. Wolf. Lame Functions. Spheroidal Wave Functions. H. Volkmer. Heun Functions. B. Sleeman & V. Kuznetsov. Painleve Transcendents. P. Clarkson. Integrals with Coalescing Saddles. M. Berry & C. Howls. Wavelets. G. Strang. 3j, 6j, 9j Symbols. L. Maximon. This list is subject to change, and all chapters are subject to editorial review and independent validation before acceptance by NIST. Contracts are in process now with some of the authors, and are impending for the others. The work is being organized and supervised by 4 NIST editors and 10 associate editors from other institutions. The NIST editors and their areas of responsibility are: D. Lozier (General), F. Olver (Mathematics), C. Clark (Scientific Applications), and R. Boisvert (Information Technology). The associate editors and their areas of responsibility are: R. Askey (special functions), M. Berry (physics), W. Gautschi (numerical analysis), L. Maximon (physics), M. Newman (combinatorics and number theory), I. Olkin (statistics), P. Paule (computer algebra), W. Reinhardt (chemistry), N. Temme (special functions), and J. Wimp (special functions). The DLMF is being modeled after the 1964 National Bureau of Standards Handbook of Mathematical Functions, M. Abramowitz and I. Stegun, editors. It is being prepared on the basis of a thorough review of the published archival literature, with emphasis on the presentation of those mathematical properties that are most useful in scientific and other applications. It will include computational information, pointers to software, illustrative applications, and graphics. It will be disseminated from a Web site at NIST with capabilities for browsing, searching, interactive visualization, and importation of information into documents or computer programs. Also, a book will be published with a CD-ROM that will reproduce many of the capabilities of the Web site. Funding has been provided by the National Science Foundation. Completion is due in 2003. Further information can be found at the project Web site, http://dlmf.nist.gov. See also http://dlmf.nist.gov/about/publications. Topic #12 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: Walter Van Assche Subject: Special Functions Posters Wolfram Research has prepared a major poster on special functions. It is divided into five distinct panels: Elliptic functions Elementary functions Hypergeometric functions Zeta and other functions Special function (general) For details and pictures of these posters one can visit http://www.specialfunctions.com Topic #13 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ 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 Article math.CA/0005095 Title: A generalization of Kummer's identity Author: Raimundas Vidunas From: Raimundas Vidunas Article math.QA/0005071 Title: The q-twisted cohomology and the q-hypergeometric function at |q|=1 Author: Yoshihiro Takeyama From: Yoshihiro Takeyama Article math.QA/0005123 Title: Refined q-trinomial coefficients and character identities Author: S. Ole Warnaar From: S. Ole Warnaar Nonlinear Sciences, abstract nlin.SI/0007001 From: Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on HyperElliptic Sigma Functions Author: Shigeki Matsutani Nonlinear Sciences, abstract nlin.SI/0005064 From: Peter Forrester Painlev\'e transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles Author: P.J. Forrester (University of Melbourne) Topic #14 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: Changes of address, WWW pages, etc. Damian G. McGuckin informs us of teh following new contact information. Address: Pacific ESI, Unit 22 8 Campbell St, Artarmon N.S.W 2064, Australia Phone: 61-2-9906-3377 Fax: 61-2-9906-3468 Email: damianm@esi.com.au Topic #15 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: About the Activity Group The SIAM Activity Group on Orthogonal Polynomials and Special Functions consists of a broad set of mathematicians, both pure and applied. The Group also includes engineers and scientists, students as well as experts. We have around 140 members scattered about in more than 20 countries. Whatever your specialty might be, we welcome your participation in this classical, and yet modern, topic. Our WWW home page is: http://math.nist.gov/opsf/ This is a convenient point of entry to all the services provided by the Group. Our Webmaster is Bonita Saunders (bonita.saunders@nist.gov). The Activity Group sponsors OP-SF NET, which is transmitted periodically by SIAM. It is provided as a free public service; membership in SIAM is not required. The OP-SF Net Editor is Martin Muldoon (muldoon@yorku.ca). To receive the OP-SF NET, send your name and email address to poly-request@siam.org. Back issues can be obtained by anonymous ftp from ftp.wins.uva.nl in the directory: pub/mathematics/reports/Analysis/koornwinder/opsfnet.dir or at the WWW addresses: http://turing.wins.uva.nl/~thk/opsfnet http://www.math.ohio-state.edu/JAT http://math.nist.gov/opsfnet/archive The NET provides fast turnaround compared to the printed Newsletter, also sponsored by the Activity Group, and edited by Renato Alvarez-Nodarse and Rafael Yanez. It appears three times a year and is mailed by SIAM. Back issues are accessible at: http://www.imn.htwk-leipzig.de/~koepf/siam.html To receive the Newsletter, you must be a member of SIAM and of the Activity Group. SIAM has several categories of membership, including low-cost categories for students and residents of developing countries. For current information on SIAM and Activity Group membership, contact: Society for Industrial and Applied Mathematics 3600 University City Science Center Philadelphia, PA 19104-2688 USA phone: +1-215-382-9800 email: service@siam.org WWW : http://www.siam.org http://www.siam.org/membership/outreachmem.htm Finally, the Activity Group operates an email discussion group, called OP-SF Talk. To subscribe, send the email message subscribe opsftalk Your Name to listproc@nist.gov. To contribute an item to the discussion, send email to opsftalk@nist.gov. The archive of all messages is accessible at: http://math.nist.gov/opsftalk/archive Topic #16 ------------ OP-SF NET 7.4 ------------- July 15, 2000 ~~~~~~~~~~~~~ From: OP-SF NET Editor Subject: Submitting contributions to OP-SF NET and Newsletter To contribute a news item to OP-SF NET, send email to poly@siam.org with a copy to the OP-SF Editor. Please note that submissions to the Net are automatically considered for the Newsletter, and vice versa, unless the contributor requests otherwise. Contributions to the OP-SF NET 7.5 should be sent by September 1, 2000. Please send your Newsletter contributions directly to the Editors: Renato Alvarez-Nodarse Departamento de Analisis Matematico Universidad de Sevilla Apdo. Postal 1160, Sevilla E-41080 Spain fax: +34-95-455-7972 e-mail: renato@gandalf.ugr.es ran@cica.es Rafael J. Yanez Departamento de Matematica Aplicada Universidad de Granada E-18071 Granada, Spain phone: +34-58-242941 fax: +34-58-242862 e-mail: ryanez@ugr.es preferably by email, and in latex format. Other formats are also acceptable and can be submitted by email, regular mail or fax. The deadline for submissions to be included in the October 2000 issue is September 15, 2000 and for the February 2001 issue it is January 15, 2001. o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o OP-SF NET is a forum of the SIAM Activity Group on Special Functions and Orthogonal Polynomials. We disseminate your contributions on anything of interest to the special functions and orthogonal polynomials community. This includes announcements of conferences, forthcoming books, new software, electronic archives, research questions, job openings. o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o Send submissions to: poly@siam.org Subscribe by mailing to: poly-request@siam.org or to: listproc@nist.gov Get back issues from URL: http://turing.wins.uva.nl/~thk/opsfnet/ WWW home page of this Activity Group: http://math.nist.gov/opsf/ Information on joining SIAM and this activity group: service@siam.org o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o The elected Officers of the Activity Group (1999-2001) are: Daniel W. Lozier, Chair Walter Van Assche, Vice Chair Charles F. Dunkl, Secretary Francisco Marcellan, Program Director The appointed officers are: Renato Alvarez-Nodarse and Rafael J. Yanez, Newsletter Editors Martin Muldoon, OP-SF NET editor Bonita Saunders, Webmaster o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o