**Subject**:**OP-SF Net Volume 6 Number 5****From**:**Daniel W Lozier <lozier@cam.nist.gov>**- Date: Wed, 22 Sep 1999 16:27:45 -0400 (EDT)
- Cc: lozier@stirling.cam.nist.gov

----- Begin Included Message ----- >From mailer@turing.siam.org Fri Sep 17 14:07:36 1999 X-UIDL: f963dd3b9c1d0ccc5ff76ff8d03fbfcb Date: Fri, 17 Sep 1999 14:07:23 -0400 (EDT) From: mailer@siam.org Subject: OP-SF Net Volume 6 Number 5 Content-Length: 81466 o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o - - - September 15, 1999 - - - - O P - S F N E T Volume 6, Number 5 - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - 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. 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 <muldoon@yorku.ca> 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 <sks@asu.edu> 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 <muldoon@yorku.ca> 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 <muldoon@yorku.ca> 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 <muldoon@yorku.ca> 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 <muldoon@yorku.ca> 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 <muldoon@yorku.ca> 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 $$ P_n(x):= c_n \int_a^b \ldots \int_a^b \prod_{j=1}^n (x-\lambda_j) \prod_{1\le i<k\le n} |\lambda_i-\lambda_k|^{2\beta} d\alpha(\lambda_1) \ldots d\alpha(\lambda_n), $$ first for $\beta=1$, next for general $\beta$. In the case \beta = 1, it is a formula valid for all orthogonal polynomials. In the case \beta \ne 1, the polynomials are no longer orthogonal. 2.12. Adam McBride One should not generalize without further motivation, for instance working on a transformation in $n$ variables that reduces to the Laplace transform if $n-1$ variables are taken zero. McBride also observed that there are two kinds of generalization. (i) Thesis-type problems that are often somewhat contrived; (ii) Real problems. We should resist generalisation for its own sake. 2.13. Tom Koornwinder The positivity result by Margit Roesler (her lecture at this workshop) should be extended to the positivity proof for the kernel in the integral representation for a Hecke-Opdam hypergeometric function (Jacobi function) associated with a root system. Similarly, the positivity of the kernel for the product formula of such functions, both in the compact and in the non-compact case, should be proved (i.e., their hypergroup property). For parameter values admitting an interpretation as spherical functions these positivity results are clear. As for quantum groups and related q-special functions, the spherical theory for quantum analogues of compact symmetric spaces is already much advanced by work of Noumi and coworkers, including Dijkhuizen and Stokman. However, the work on the spherical theory for quantum analogues of non-compact Riemannian symmetric spaces has just started (see lectures by Koelink and Stokman in this workshop), while even less work has been done on the case that $q$ is on the unit circle, but not necessarily a root of 1 (analytically very interesting and challenging). 2.14. Norman Wildberger Give explicit descriptions of representations of groups. Koelink adds that a lot in this direction can already be found in the 3-volume work "Representation of Lie groups and special functions" by N.J. Vilenkin and A.U. Klimyk. 3. Mathematical physics 3.1. Luc Vinet Solvable modes are important and lead to special functions. In physics it is very important to explain phenomena by symmetry. More people should be working on the factorial n problem. Find a physical interpretation for the generalized Rogers-Ramanujan identity pointed out by Dennis Stanton (his lecture at this workshop), see also the preprint "Variants of the Rogers-Ramanujan identities" by T. Garrett, M.E.H. Ismail and D. Stanton, http://www.math.umn.edu/~stanton/pap.html. There are many open problems for special functions in black hole physics and in string theory. There is lots to explain in the interplay between mathematical physics and algebraic combinatorics. Finally study special function solutions of non-linear equations (Painleve theory). 3.2. Frank Olver The project "Digital Library of Mathematical Functions" (see Lozier's lecture in this workshop) will be very important for bringing special functions to physicists and vice versa, and also for drawing attention to research needs of practical importance. There have been an enormous, and increasing, number of citations to the "Handbook of mathematical functions", Abramowitz and Stegun (eds.) in the Science Citation Index, and the majority are in physics papers. Koelink asked if a similar citation analysis has been made for the Bateman project. Olver is not aware of such an analysis. 3.3. Jesus Sanchez Dehesa Study entropy integrals (see Dehesa's lecture in this workshop). These are important in physical applications and much of the theory still has to be developed. Orthogonal polynomials in several variables are also relevant for physics and in connection with numerical problems. Study the asymptotics of L^p norms of sequences of orthogonal polynomials. We know only the dominant terms. Finally there is an important connection between Information theory and Special functions. In closing the discussion, Dunkl suggested that people send e-mail to the organizers or to opsftalk with other suggestions. In correspondence arising in connection with this Report, Andre Ronveaux mentions that "Differential Equations problems coming from Separation of variables in classical linear PDE of Mathematical Physics are not yet completely investigated, even in dimension 3.(I am thinking mainly on Spheroidal and Ellipsoidal coordinate systems involving many eigenvalue problems, connection between solutions...)". Topic #8 ------------ OP-SF NET 6.5 ----------- September 15, 1999 ~~~~~~~~~~~~~ From: Richard Askey <askey@math.wisc.edu> 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 <vandiej@uchile.cl> 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 <koepf@imn.htwk-leipzig.de> 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 <Walter.VanAssche@wis.kuleuven.ac.be> 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 <muldoon@yorku.ca> 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 <lozier@cam.nist.gov> 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 <bogardo@siam.org> 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 <bogardo@siam.org> 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 <bogardo@siam.org> 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 <muldoon@yorku.ca> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <koekoek@twi.tudelft.nl> 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 <y.chen@ic.ac.uk> 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 <felder@math.ethz.ch> 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 <bdt@math.wayne.edu> 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 <meh@nadn.navy.mil> 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 <schubert@lapp.in2p3.fr> 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 <helmut@cam.wits.ac.za> 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 <thk@wins.uva.nl> 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 <vadim@amsta.leeds.ac.uk> 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 <terasoma@mpim-bonn.mpg.de> 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 <jakobsen@math.ku.dk> 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 ~~~~~~~~~~~~~~~ From: OP-SF NET Editor <muldoon@yorku.ca> Subject: Changes of Address, WWW Pages, etc Jan Felipe van Diejen has informed us of a change of address of his home page. 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