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o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o - - - May 17, 1995 - - O P - S F N E T Volume 2, Number 3 - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - - Tom H. Koornwinder, Editor thk@fwi.uva.nl - - - - - - The Electronic News Net of the SIAM Activity Group - - on Orthogonal Polynomials and Special Functions - - - - Please send contributions to: poly@siam.org - - & address changes to: poly-request@siam.org - - - o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o Today's Topics: 1. Newsletter Editor announces his resignation 2. Position opening for Newsletter Editor 3. Nominating Committee 4. On the Newsletter and its temporary substitute 5. Conference in honour of Lee Lorch 6. Mini-Program on Special Functions, q-Series and Related Topics 7. Minisymposium at ICIAM '95 (Hamburg) 8. Minisymposium at SIAM Annual Meeting in Charlotte 9. Minisemester on Quantum groups and quantum spaces 10. Monroe Martin Prize 11. Gautschi Festschrift 12. New book by Mathai and Provost 13. New book on mathematical software 14. Short book announcements 15. New errata for the book Basic hypergeometric series 16. Seminaire Lotharingien de Combinatoire 17. AMS preprint server 18. BibNet - A Bibliography Network Project 19. Basic options available on AT-NET 20. ftp and WWW addresses 21. Changes of addresses 22. Recent additions to ftp site on orthogonal polynomials and related special functions 23. Obtaining back issues of OP-SF Net and the Newsletter 24. Abstract of Spigler's talk given at San Diego minisymposium 25. Gabor Szego - One hundred years (long topic) 26. Remarks on Problem 7, The incomplete Airy function (latex file) Calendar of events: see issue/topic: 1995 May 29-June 2: Formal Power Series and Combinatorics, Paris 1.8 #13 June 9-10: Conference in honour of Lee Lorch 2.2 #4 June 12-23: Fields Institute mini-program on Special functions etc. 2.2 #5 June 22-24: q-Special Functions at Prague Quantum Groups meeting 2.1 #4 July 4, 15.30 hour: Minisymposium at ICIAM '95, Hamburg: "Orthogonal Polynomials and Spectral Methods" 2.3 #7 July 10-16: Conference on Symmetry Methods in Physics 2.2 #6 July 19-21: First Maghrebian Colloquium on Analysis 1.9 #4 October 23-26: Minisymposium at SIAM Annual Meeting in Charlotte 2.3 #8 November 6- December 1: Minisemester on Quantum groups and quantum spaces 2.3 #9 1996 May 6-26: CRM Workshop on the Theory of Special Functions 2.1 #5 Topic #1 ----------------- OP-SF NET ------------------ May 17, 1995 From: Charles F. Dunkl, Chair of Activity Group <cfd5z@fermi.clas.virginia.edu> Subject: Newsletter Editor announces his resignation We are sorry to announce the resignation of Eugene Tomer as Newsletter Editor. We have all benefitted from his valuable contributions to the group. Throughout his tenure, from 1992 to now the Newsletter has been of consistent high quality in both content and design, and we express our gratitude to Eugene. Topic #2 ----------------- OP-SF NET ------------------ May 17, 1995 From: Charles F. Dunkl, Chair of Activity Group <cfd5z@fermi.clas.virginia.edu> Subject: Position opening for Newsletter Editor The group is looking for a person willing to produce 2 or 3 issues of the newsletter per year. We can be very flexible about the design. The position involves cooperation with the electronic OP-SF Net editor (currently Tom Koornwinder) in the collection of news items (so a good e-mail connection is vital). At present SIAM HQ in Philadelphia receives the master draft, makes copies and does the mailings. Reimbursement is available for expenses such as materials, mailings, Fedex costs, etc. (but SIAM does not pay travel costs to meetings, just for the record). Persons interested in this task or who would just like more information are asked to get in touch with George Gasper (george@math.nwu.edu) or Charles Dunkl (cfd5z@virginia.edu). It may well happen that only one more issue will be produced in 1995. Topic #3 ----------------- OP-SF NET ------------------ May 17, 1995 From: Charles F. Dunkl, Chair of Activity Group <cfd5z@fermi.clas.virginia.edu> Subject: Nominating Committee The terms of the present elected officers expire at the end of 1995. The president of SIAM, Margaret Wright, has appointed Richard Askey and Walter van Assche to serve on the Nominating Committee together with the elected officers (who serve ex officio). Any input on this matter from members is welcome. Please contact Dick Askey <askey@math.wisc.edu> or Walter Van Assche <Walter.VanAssche@wis.kuleuven.ac.be>. Topic #4 ----------------- OP-SF NET ------------------ May 17, 1995 From: Tom Koornwinder (OP-SF Net editor) <thk@fwi.uva.nl> Subject: On the Newsletter and its temporary substitute The last issue of the Newsletter which will have reached you is the Winter 1994 issue (Volume 5, Number 2). The Spring 1995 issue (due in March 1995) has not appeared. Because of Eugene Tomer's resignation as an editor of the Newsletter (see Topic #1), the Summer 1995 issue (due in June 1995) will not appear either, at least not in its usual form. Normal appearance of the Newsletter, possibly with lower frequency, has to wait until a new editor has been found (see Topic #2). In order to compensate for this, the officers of the Activity Group have decided to take the following emergency measures: 1. Items which used to be only included in the Newsletter and not in OP-SF Net (because they are longer articles or because they contain formulas and therefore have to be transmitted in TeX form) will now be included in OP-SF Net, but at the end of each issue. 2. The items mentioned in 1. will be restricted to urgent ones and to reactions to earlier contributions to the Newsletter. The problem section will be temporarily discontinued, but reactions to earlier posed problems will be included in OP-SF Net. Research questions will also be included, in general. 3. A selection from recent issues of OP-SF Net will be regularly sent in printed form to members who do not subscribe to OP-SF Net. This will be usually be a straightforward printing of a plain text file, without TeX typesetting. It is our intention to prepare a first mailing (with selection from OP-SF Net 1.9, 2.1, 2.2 and 2.3, i.e. starting from the December 1994 issue) very soon. Tentatively, future hardcopy mailings will be made out of two consecutive issues of OP-SF Net (the frequency of OP-SF Net is once every 6 or 8 weeks). 4. We strongly recommend members with email address who do not yet subscribe to OP-SF Net, to do so from now on (send mail to poly-request@siam.org). If you do not like to read an electronic newsletter from your screen then you may print it yourself, and thus have the information sooner. 5. If you are mailing contributions to OP-SF Net which would have been normally meant for the Newsletter, please put it in the following form. If there are no formulas or very few ones, please contribute your material in plain text, not as a TeX input file. Avoid diacritical signs as much as possible. If they are necessary in order to avoid confusion, please give these signs in TeX notation. Also you may locally use some pseudo-TeX for minor formulas. Contributions in which formulas are crucial may be given as TeX input files. Use your favourite TeX dialect( plain TeX, latex, AMS-TeX or amslatex), but avoid separate input files for macros. 6. Send all contributions by email to poly@siam.org. We hope the old routines can be taken up again soon. Topic #5 ----------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net Editor <thk@fwi.uva.nl> Subject: Conference in honour of Lee Lorch As was announced earlier in OP-SF Net 1.7, topic 8, a conference on "Special Functions and Related Topics in Analysis" will be held at York University, North York (Metropolitan Toronto), Ontario, Canada on Friday and Saturday, June 9 - 10, 1995. It will be dedicated to Lee Lorch on the occasion of his forthcoming 80th birthday (September 20, 1995). Updated information on this conference can be obtained on the World Wide Web: http://www.math.yorku.ca/Conferences/LL80/menu.html or, of course, by e-mailing muldoon@mathstat.yorku.ca. From this source, I have taken the following preliminary list of titles. Mark Ashbaugh (University of Missouri), "Properties of Bessel Function Zeros Related to Isoperimetric Inequalities for Eigenvalues of Membranes and Plates" Richard Askey (University of Wisconsin), "Bessel functions and how to use them when considering more general classes of functions" Chandler Davis (University of Toronto), "An agitator, or what?" James A. Donaldson (Howard University), "The Linear Shallow Water Theory and Dirichlet-Neumann Operators" Arpad Elbert (Hungarian Academy of Sciences), "Eigenvalues for half-linear differential equations" Mary Gray (American University), "Including the Excluded: Lee Lorch's Contributions to Expanding Access to Mathematics". Johnny Houston (National Association of Mathematicians), TBA Mourad Ismail (University of South Florida), "Integral operators and q-Sturm-Liouville problems". Jean-Pierre Kahane (Universite de Paris - Orsay), "Summability, order and products of Dirichlet series" Amram Meir (York University), "Degree distribution in random trees" A. McD. Mercer (University of Guelph), "On 'deleting a row and column' from a differential operator" Angelo Mingarelli (Carleton University), TBA Donald J. Newman (Temple University), "The Morley triangle" P.G. Rooney (University of Toronto), "On the Hankel transformation" Mark Pinsky (Northwestern University), "Pointwise convergence of the Fourier integral and related orthogonal expansions in several variables" Dennis Russell (York University), "Convolution of sequences, and application of Fourier transforms to the solution of some problems in linear inequalities" Cora Sadosky (Howard University), "Restricted BMO in product spaces" Walter Van Assche (Katholieke Universiteit Leuven), "Zeros of orthogonal polynomials and eigenvalues of matrices" Roderick Wong (University of Manitoba and City University of Hong Kong), "Asymptotics and Special Functions" Abstracts of most of these talks can be obtained from the WWW site. Topic #6 ----------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net Editor <thk@fwi.uva.nl> Subject: Mini-Program on Special Functions, q-Series and Related Topics The Mini-Program on Special Functions, q-Series and Related Topics will be held during 12-23 June, 1995 at the University of Toronto. It is an activity of the Fields Institute. See OP-SF Net 2.1, Topic #3 and 2.2, Topic #5 for details. The following additional information is available now: 1. Sergei Suslov has been added to the list of invited speakers during the second week. 2. It is still possible to apply for giving a contributed lecture during the second week (if the schedule has place). For all further information: Lynne Wight The Fields Institute, 185 Columbia Street West Waterloo, Ontario, Canada N2L 5Z5 e-mail: spfunct@fields.uwaterloo.ca Fax: (519) 725-0704 Phone: (519) 725-0096 Topic #7 ----------------- OP-SF NET ------------------ May 17, 1995 From: Martin Muldoon <muldoon@mathstat.yorku.ca> Subject: Minisymposium at ICIAM '95 (Hamburg) In OP-SF Net 1.8, Topic #16 the program was given for the Minisymposium at ICIAM '95 (Hamburg, Germany, July 3-7, 1995) co-sponsored by the SIAM Activity Group on Orthogonal Polynomials and Special Functions. Two of the announced speakers, Marcel de Bruin and Walter Van Assche, had to withdraw from the program because of other obligations. Instead of them, Bernie Shizgal, and Andre/Paco have agreed to speak, so there are still four talks. Below follows the revised program. "Orthogonal Polynomials and Spectral Methods" A Minisymposium sponsored by the SIAM Activity Group on Orthogonal Polynomials and Special Functions to be held during ICIAM '95, Hamburg, Germany, July 3-7, 1995 Organizers: Francisco Marcellan (EPS Carlos III, Madrid, Spain) (pacomarc@ing.uc3m.es) and Andre Ronveaux (Facultes Universitaires N.D. de la Paix, Namur, Belgium) (Andre.Ronveaux@fundp.ac.be) Date: Tuesday, July 4, 1995. Time: 15:30 Speakers: Francisco Marcellan (EPS Carlos III, Madrid, Spain) and Andre Ronveaux (Facultes Universitaires N.D. de la Paix, Namur, Belgium): "A short introduction to polynomials orthogonal with respect to Sobolev inner products" Arieh Iserles (University of Cambridge, England): "Sobolev-orthogonal polynomials and pseudospectral methods for parabolic PDE's" Yvon Maday (Universite Paris VI, France): "Spectral Methods for solving axisymmetric partial differential equations". Bernie D. Shizgal (University of British Columbia, Canada): "The quadrature discretization method (QDM) in the solution of the Schroedinger equation with non-classical basis functions". Topic #8 ----------------- OP-SF NET ------------------ May 17, 1995 From: Martin Muldoon (Program Director) <muldoon@mathstat.yorku.ca> Subject: Minisymposium at SIAM Annual Meeting in Charlotte Subject to approval of the Organizing Committee, the following minisymposium, sponsored by our Activity Group, will take place at the SIAM Annual Meeting, Charlotte, North Carolina, October 23-26, 1995. Minisymposium Title: Computational Aspects of Special Functions and Orthogonal Polynomials Organizer: Martin E. Muldoon (Program Director, SIAM Activity Group on Orthogonal Polynomials and Special Functions) Affiliation: York University, North York, Ontario, Canada Scope of minisymposium: Many problems of applied mathematics lead to approximations which eventually involve the computation of special functions. Such approximations can provide insight as well as computational benefits. The important issues of software availability and reliability are addressed in general terms in the talk by Lozier and in some specific cases in that by Temme. The talks by Dunkl and Boyd address specific problems related to the computation of special functions form surface measures of ellipsoids and in solitary wave theory respectively. Temme and Boyd address the issue of asymptotic behavior (i.e., the situation which arises when a parameter is large). There has been much recent work on classes of orthogonal polynomials and the relations between the orthogonality measures, recurrence relations and zeros of the polynomials. The talk by Gautschi will be concerned with some numerical aspects of these relations. Speakers: John P. Boyd (University of Michigan, Ann Arbor, MI), "Numerical verification of exponential asymptotics via a Chebyshev polynomial pseudospectral algorithm in multiple precision: radiation coefficient of a weakly nonlocal solitary wave" Charles F. Dunkl (University of Virginia, Charlottesville, VA), "Hyperelliptic integrals, the surface measure of ellipsoids, and response surfaces" (with Donald E. Ramirez) Walter Gautschi (Purdue University, West Lafayette, IN), "Computing orthogonal polynomials of Sobolev type" Daniel W. Lozier (National Institute of Standards and Technology, Gaithersburg, MD), "Software issues in the computation of special functions" Nico M. Temme (CWI, Amsterdam), "Large parameter evaluations of some classical distribution functions" Topic #9 ----------------- OP-SF NET ------------------ May 17, 1995 From: Stanislaw Zakrzewski <szakrz@fuw.edu.pl> Subject: Minisemester on Quantum groups and quantum spaces The Stefan Banach International Mathematical Center in Warsaw, Poland will host a Minisemester on "Quantum groups and quantum spaces" during November 6 - December 1, 1995. The last (fourth) week will also include applications to the theory of special functions. The program for the four weeks is as follows: 1st week: Topological aspects of quantum groups (C*-algebra approach); 2st week: Algebraic aspects (Hopf algebra approach); 3rd week: Noncommutative geometry, Semiclassical limits; 4th week: Applications to physical models and to the theory of special functions. Participants are expected to come, as a rule, for one week of their choice (optionally, two weeks). The contact person is: Witold Kondracki <witekkon@impan.gov.pl> Inst. of Math. Polish Acad. Sci. Sniadeckich 8, 00-950 Warszawa, Poland. Information by email can also be obtained from Stanislaw Zakrzewski <szakrz@fuw.edu.pl> Head of the Local Organizing Committee is S.L. Woronowicz (Warsaw Univ). Topic #10 ----------------- OP-SF NET ------------------ May 17, 1995 From: Frank W.J. Olver <olver@Glue.umd.edu> Subject: Monroe Martin Prize The Institute for Physical Science and Technology at the University of Maryland, College Park is pleased to announce the Fifth Monroe Martin Prize. The prize will be awarded for an outstanding paper in applied mathematics (including numerical analysis) by a young research worker. Candidates must be residents of North America and not more than 35 years of age at the filing deadline. The submitted paper must be by a single author and have been published, or accepted for publication, in the open literature. The work must not have been performed in connection with the completion of requirements for an academic degree, nor may it have been performed by an author associated with the University of Maryland. Applications from qualified candidates, or nominations, are solicited for the Monroe Martin prize. Entries should include a copy of the paper or contribution, with a covering letter, and for full consideration be submitted on or before July 31, 1995 to J. A. Yorke, director Institute for Physical Science and Technology University of Maryland College park, Maryland 20742, USA The award will be announced by November 1, 1995. The recipient will be asked to present his or her work at the Monroe Martin Lecture at the University of Maryland in December, 1995, and will be awarded a prize of $2,000 plus travel expenses. The Monroe Martin prize was established to commemorate the achievements of Professor Monroe Martin, former Director of the Institute for Fluid Dynamics and Applied Mathematics, and Chair of the Mathematics Department at the University of Maryland. Previous prize winners are Neil Berger, Marshall Slemrod, Jonathan Goodman, and Marek Rychlik. Topic #11 ---------------- OP-SF NET ------------------ May 17, 1995 From: Martin Muldoon <muldoon@mathstat.yorku.ca> Subject: Gautschi Festschrift The following book appeared: R. V. M. Zahar, ed., Approximation and Computation: A Festschrift in Honor of Walter Gautschi, International Series of Numerical Mathematics, vol. 119, Birkhauser, Boston-Basel-Berlin, 1994. This book contains the proceedings of an International Symposium on "Special Functions, Approximation, Numerical Quadrature and Orthogonal Polynomials", held at Purdue University in December 1993, in celebration of the 65th birthday of Walter Gautschi. It includes 37 invited papers by 60 authors from 16 countries as well as a fascinating autobiographical article by Gautschi entitled "Reflections and Recollections". A list of Gautschi's publications and Ph.D. students is included. The book is carefully edited with a uniform appearance and very complete subject and author indices. Topic #12 ---------------- OP-SF NET ------------------ May 17, 1995 From: Hans Haubold <haubold@vs3.kph.tuwien.ac.at> Subject: New book by Mathai and Provost Authors: A.M.Mathai and S.B. Provost Title: Quadratic Forms in Random Variables: Theory and Applications Publisher: Marcel Dekker, Inc., New York, 1992 Statistics: textbooks and monographs volume 126 The monograph discusses quadratic forms and second degree polynomials in real random variables and normal random vectors and matrices. Naturally, special functions appear particularly in chapter 4 as they are used as representations of densities and distribution functions of quadratic forms. The monograph offers a compilation of known and previously unpublished results in this field, enriched by exercises given in all chapters and can also be used as a reference tool for theoretical and applied statisticians as well as applied mathematicians. The book contains seven chapters: (1) Preliminaries, (2) Quadratic Forms in Real Variables, (3) Quadratic Forms in Random Variables, (4) The distribution of Quadratic Forms, (5) Chi-squaredness and Independence, (6) Generalized Quadratic Forms, and (7) Applications. Topic #13 ----------------- OP-SF NET ------------------ May 17, 1995 From: Hans Haubold <haubold@vs3.kph.tuwien.ac.at> Subject: New book on mathematical software Authors: T. Watanabe, M. Natoni, and T. Oguni (Editors) Title: Mathematical Software for the P.C. and Workstations: A Collection of Fortran 77 Programs Publisher:Elsevier Science B.V., North-Holland, Amsterdam, 1994 The book is the translation of its Japanese version from 1989 and contains instructions and mathematical outline for numerical computations focusing on (1) the Gamma function and related functions, Bessel functions of first and second kind and related functions, various orthogonal polynomials, exponential and trigonometric integrals, the error function and related functions, the Fresnel integrals, the complete elliptical integrals and random number generation, (2) algebraic equations, transcendental equations, ordinary differential equations, simultaneous linear equations, inverse matrices, eigenvalue problems of matrices, and (3) data analysis, including least-square methods, optimization methods, interpolations, and Fourier transforms. A diskette is included with all the source code in double precision for real values of the argument of the function. The bibliographies attached to each chapter in the book refer particularly to Japanese research work done in the field of numerical analysis and supplements in this sense D.W. Lozier and F.W.J. Olver's very useful paper that assists to locate approximations and software for the numerical generation of special functions. Topic #14 ---------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: Short book announcements The following books appeared: I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Second edition, 1995. Compared to the first 1979 editions two new chapters have been added. In particular, Macdonald's polynomials associated with root system A_n are treated here. N. Ja. Vilenkin and A.U. Klimyk, Representation of Lie groups and special functions, Recent Advances, Kluwer, 1995. This book, a sequel to the three-volume series by the same two authors which appeared during the past few years, starts with a treatment of Dunkl operators and of Heckman-Opdam hypergeometric functions. G. Heckman and H. Schlichtkrull, Harmonic analysis and special functions on symmetric spaces, Perspectives in Mathematics 16, Academic Press, 1994. This book contains the notes by the two authors of courses delivered at sessions of the European School of Group Theory in Luminy, France, 1991 and Twente, Netherlands, 1992. In particular, the two contributions by Heckman deal with aspects of special functions: pp. 1-89, Hypergeometric and spherical functions pp. 189-208, Are K-invariant Eisenstein integrals for G/H hypergeometric functions? V. Baldoni and M.A. Picardello (eds.), Representations of Lie groups and quantum groups, Pitman Research Notes in Mathematics Series 311, Longman Scientific & Technical, 1994. This book contains the notes of courses delivered at the session of the European School of Group Theory in Trento, Italy, 1992. The second half of the contribution by Tom Koornwinder contains a tutorial on q-hypergeometric functions and related orthogonal polynomials: pp. 46-128, Compact quantum groups and q-special functions. J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford University Press, 1994. The symmetric cones provide a good setting for zonal polynomials and hypergeometric functions of matrix argument. Such functions were earlier studied in multivariate statistics, cf. the work by A.T. James and others. W. C. Connett, M.-O. Gebuhrer and A. L. Schwartz (eds.), Applications of hypergroups and related measure algebras, Contemporary Mathematics 183, American Math. Soc., 1995. These are the refereed Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on "Applications of hypergroups and related measure algebras", Seattle, Washington, USA, Augustus 1993. A lot of hypergroups related special functions stuff can be found in it. Topic #15 ----------------- OP-SF NET ------------------ May 17, 1995 From: George Gasper <george@math.nwu.edu> Subject: New errata for the book Basic hypergeometric series Errata, updates of the References, etc., (as of March 30 27, 1995) for the book Basic Hypergeometric Series, by George Gasper and Mizan Rahman, Encyclopedia of Mathematics and its Applications, Vol. 35, Cambridge University Press, Cambridge-New York, 1990 are now available on file from WWW: http://www.math.nwu.edu/preprints/gasper/index.html Below are the errata added to the list since December 27, 1994: p. 54, Ex. 2.21: Replace `` $({aq \over \lambda})^n$ '' by `` $({a^2q^2 \over efgh})^n$ ''. p. 62, line below (3.2.9): Replace ``min'' by ``max'' . p. 95, Ex. 3.21: In the numerator on the right side of the equation replace `` $(A, AQ/B; Q)_n$ '' by `` $(Q, AQ/B; Q)_n$ '' . p. 147, eq. (6.4.11): In the numerator of the first ${}_{10} \phi_9$ on the right side of the equation replace `` $\nu_1, \sqrt \nu_1$ '' by `` $\nu_1, q \sqrt \nu_1$ '' . p. 267, lines 11--12 up: Replace `` $U_q( \quad , $ orthogonal '' by `` $U_q({\it sl}(2)), \, q$-orthogonal '' . Topic #16 ----------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: Seminaire Lotharingien de Combinatoire The Seminaire Lotharingien de Combinatoire has existed for fourteen years as a joint seminar in Combinatorics for the Universities of Bayreuth, Erlangen and Strasbourg, as well as the other Universities along the Lotharingian boundaries. The proceedings of these meetings were regularly published by the University of Strasbourg, but they were not widely available in libraries. Quite a lot of papers have appeared in these Proceedings which dealt with orthogonal polynomials and special functions, and their interpretation in combinatorics. Many papers from these earlier Proceedings can now be obtained via the internet (World Wide Web): http://cartan.u-strasbg.fr/~slc Form the information page there I pass you the following information. Regular meetings will continue in the same spirit, but it has been decided that the Seminar will now host a new international electronic journal carrying the same name. The main feature is the creation of two new sections: one that welcomes original mathematical papers duly refereed; the other devoted to papers of general interest that will be the state of the art in a specific subject. The proceedings of the periodical seminars (the 33rd was held last September 1994) will be published in the third section. Finally, whenever possible, papers that appeared in the previous (paper) issues of the I.R.M.A. Strasbourg collection will become electronically available in the fourth section. We are starting with an editorial board that reflects our European initiative. Most of its members participate in a European program in Algebraic Combinatorics. We mean to enlarge the editorial board to scholars from the entire world showing a definite interest in the very many facets of Combinatorics. Our intention is to collaborate with the Electronic Journal of Combinatorics published in America under the auspices of Herb S. Wilf, who is also a member of our editorial board. Editors: Dominique Foata <foata@math.u-strasbg.fr> Adalbert Kerber <kerber@btm2x6.mat.uni-bayreuth.de> Volker Strehl <strehl@informatik.uni-erlangen.de> Topic #17 ---------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: AMS preprint server Recently, the AMS has started with an electronic preprint server. It carries preprints in all areas of mathematics. These preprints do not need to be physically present at the server. One may contribute preprints by just sending an abstract and a pointer to a place at the internet where the TeX file is residing. The AMS server will then automatically connect interested browsers across the Internet to the appropriate server where the preprint resides. One of the search possibilities at the AMS server is by Mathematics Subject Classification. It turns out that, until now, nothing has been contributed with classification 33-XX (Special functions). Also, 41-XX (Approximations and expansions) and 42-XX (Fourier analysis) do not yet carry any material on orthogonal polynomials. I want to recommend you to send abstracts of your recent and future preprints to this AMS server, even if you will put the fiel itself at another server (for instance the one supervised by Waleed Al-Salam). You will find more information in Notices AMS 42, Number 5, May 1995, p.583. The server is accessible on the World Wide Web at the address http://e-math.ams.org/web/preprints/ or via the AMS e-MATH home page http://e-math.ams.org Topic #18 ---------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: BibNet - A Bibliography Network Project The March 1995 issue of SIAM News mentions at p.4 a new bibliographical resource for the field of scientific computing. BibNet is a public-domain bibliography archive in BibTeX format. The archive contains three types of bibliographies: by author (as in a curriculum vitae), by subject (as in a review article), and by institution (for example, the tech-reports of a university department). This initiative is a step toward sharing information electronically, and it allows scientists to: provide complete and updated information on their own work, find information about ongoing research or specific topics, have an efficient pointer to electronically accessible material, and simplify the work of preparing publications. The BibNet archive is maintained at the University of Utah, on ftp.math.utah.edu, where the files are stored in /pub/bibnet. It is mirrored nightly from there to a few other Internet archive sites. BibNet is accessible through anonymous ftp, Netlib, Mosaic, Gopher and e-mail. Readers can search the BibNet database or download individual author bibliographies by directing their favorite Web browsers to http://netlib.att.com/netlib/bibnet.html In the BibNet databese, try for instance the keyword hypergeometric. Topic #19 ----------------- OP-SF NET ------------------ May 17, 1995 From: editors of AT-NET <approx@math.technion.ac.il> Subject: Basic options available on AT-NET For the convenience of the subscribers, and since we have recently gone over to a new machine, we include a review of a few basic options available on AT-NET. A) To submit a message to AT-NET, send it to: approx@math.technion.ac.il or at-net@math.technion.ac.il B) To get the list of subscribers to AT-NET, send an e-mail message with body: review at-net to the address: listproc@math.technion.ac.il C) You have various options on how to obtain the files of previous Bulletins. a) An index of bulletins (as of January 95) is obtainable by posting the message index at-net to listproc@math.technion.ac.il Bulletins are indexed by year and month. One can download, after one knows the correct format of the name using the command get filename addressed to the same address. b) By lynx, xmosaic, etc..... to http://gauss.technion.ac.il/lists/at-net c) By ftp to gauss.technion.ac.il anonymous login, with password your address. Do cd lists/at-net to get into the correct subdirectory, and then the usual downloading commands. d) All previous Bulletins are presently archived at the home page of JAT, http://math.ohio-state.edu/Groups/JAT D) To sign on to AT-NET, do one of the following from the e-mail address you wish to have all messages sent to. Send an e-mail message to: listproc@math.technion.ac.il with the body of the message consisting of the one line: sub at-net <given_name> <surname> Your subscription should be automatically accepted. You will receive notice of this fact, along with additional information. If you have problems, write to Allan Pinkus at approx@math.technion.ac.il or Carl de Boor at atnet@stolp.cs.wisc.edu Topic #20 ----------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: ftp and WWW addresses Here follows an extension and correction of the list ftp and WWW addresses relevant for our field, which was started in OP-SF Net 2.2, Topic #11. Please let me know about further addresses. I am also interested in hearing about WWW home pages and anonymous ftp sites of individuals. Below a (*) will mean that the address was not yet included in the previous issue or is a correction of the address given there. Organisations: AMS e-MATH homepage: WWW: http://e-math.ams.org (*) AMS preprint server: WWW: http://e-math.ams.org/web/preprints/ AT-NET: WWW: http://gauss.technion.ac.il/lists/at-net (*) Bibnet search: WWW: http://netlib.att.com/netlib/bibnet.html Cern (Geneva): WWW: http://www.cern.ch Computer Algebra Nederland (CAN): WWW: http://www.can.nl Electronic Journal of Combinatorics: WWW: http://ejc.math.gatech.edu:8080/Journal/journalhome.html Guide to Available Mathematical Software: WWW: http://gams.nist.gov hep-th (theoretical high energy physics): email: hep-th@xxx.lanl.gov or hep-th@babbage.sissa.it WWW: http://xxx.lanl.gov/hep-th ICIAM 95, Hamburg: WWW: http://ganymed.mt2.tu-harburg.de/~jo/ Institute for Math. and its Appl., Minneapolis: WWW: http://www.ima.umn.edu Journal of Approximation Theory: WWW: http://www.math.ohio-state.edu/JAT Lee Lorch Conference: WWW: http://www.math.yorku.ca/Conferences/LL80/menu.html Mathematics Information Servers: WWW: http://www.math.psu.edu/OtherMath.html (*) Netlib: WWW: http://netlib.att.com/netlib (*) Netlib Search: http://netlib.att.com/netlib/search.html Newsgroup sci.math: WWW: news:sci.math Newsgroup sci.math.research: WWW: news:sci.math.research Orthogonal polynomials and related special functions (W. Al-Salam): ftp: euler.math.ualberta.ca q-alg (quantum algebra including knot theory): email: q-alg@eprints.math.duke.edu WWW: http://www.msri.org/preprints (*) Seminaire Lotharingien de Combinatoire WWW: http://cartan.u-strasbg.fr/~slc SIAM: WWW: http://www.siam.org gopher: gopher.siam.org ftp: ae.siam.org Tables of contents of math journals: WWW: gopher://gopher.cecm.sfu.ca/11/Resources WWW virtual library, mathematics WWW: http://euclid.math.fsu.edu/Science/math.html Individuals: Peter Borwein WWW: http://www.cecm.sfu.ca/personal/pborwein/Homepage.html (*) George Gasper WWW: http://www.math.nwu.edu/preprints/gasper/index.html Tom Koornwinder: ftp: ftp.fwi.uva.nl, in directory pub/mathematics/reports/Analysis/koornwinder (*) Daniel Loeb: WWW: http://www.labri.u-bordeaux.fr/~loeb (*) Martin E. Muldoon: WWW: http://www.math.yorku.ca/Who/Faculty/Muldoon/menu.html Paul Nevai: WWW: http://www.math.ohio-state.edu/~nevai Herbert Wilf: WWW: http://www.cis.upenn.edu:80/~wilf Doron Zeilberger: Ftp: ftp math.temple.edu, dir pub/zeilberg WWW: http://www.math.temple.edu/~zeilberg Topic #21 ---------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: Changes of addresses M. Noumi and M.S. Dijkhuizen, both working on quantum groups and q-special functions, moved in Japan from Tokyo to Kobe. Their new address is: Masatoshi Noumi and Mathijs S. Dijkhuizen Department of Mathematics Faculty of Science Kobe University Rokko, Kobe 657 Japan fax: +81-78-803 0723 phone (secretary): +81-78-803 0515 phone (Noumi's office): +81-78-803 0505 email Noumi: noumi@math.s.kobe-u.ac.jp email Dijkhuizen: msdz@math.s.kobe-u.ac.jp J.F. van Diejen, who is working on q-difference operators related to multivariable analogues of Askey-Wilson polynomials, has moved from Amsterdam to Tokyo. His new address is: Jan Felipe van Diejen Department of Mathematical Sciences The University of Tokyo 7-3-1 Hongo, Bunkyo-ku Tokyo 113 Japan phone: +81-3-3812 2111 ext. 4078 fax: +81-3-3814 9488 email: jand@math.s.u-tokyo.ac.jp Topic #22 ---------------- OP-SF NET ------------------ May 17, 1995 From: Waleed A. Al-Salam <waleed@euler.math.ualberta.ca> Subject: Recent additions to ftp site on orthogonal polynomials and related special functions Waleed Al-Salam has an anonymous ftp site at euler.math.ualberta.ca on orthogonal polynomials and related special functions. It can also be accessed by using World Wide Web (xmosaic or Mosaic) at the address ftp://euler.math.ualberta.ca. Manuscripts can be obtained from directory pub and subdirectories. In particular, examine the CONTENTS file in the pub directory. If you wish to receive periodic info regarding recent additions to this database please send an email message to waleed@euler.math.ualberta.ca with your e-mail address. You are invited to submit one or more of your not-yet-in-print manuscripts which you wish to make available to the general OPS community. They should be in Plain TeX, LaTeX, AMSTeX or AMSLaTeX format. These manuscripts can be submitted by one of the following methods: a) E-mail to waleed@EULER.MATH.UALBERTA.CA b) anonymous ftp to euler.math.ualberta.ca then "cd submissions" and finally "put file" where "file" is the name of the file you wish to deposit. Between March 25, 1995 and May 15, 1995 the following additions were put in the pub directory: D. R. Masson, Polynomial asymptotic and spectral properties. (see /ftp/pub/ masson3.tex). G. Gasper and W. Trebels, Ultraspherical multipliers revisited. (see /ftp/pub/gasper/qtleindler.tex) G. Gasper and W. Trebels, A Riemann-Lebesgue lemma for Jacobi expansion. (see /ftp/pub/gasper/qtbutzer.tex). M. Noumi, Macdonald's symmetric polynomials as zonal spherical functions. (see /ftp/pub/noumi.tex). M. Noumi and T. Sugitani, Quantum symmetric spaces and related q-orthogonal polynomials. (see /ftp/pub/noumi-sugitani.tex) J. F. van Diejen, On the diagonalization of difference Calogero-Sutherland systems. (see /ftp/pub/diejen.ps) R. Floreanini and L. Vinet, A model for the continuous q-ultraspherical polynomials. (see /ftp/pub/flor.tex) R. Floreanini, J. LeTourneux and L. Vinet, More on the q-oscillator algebra and q-orthogonal polynomials. (see /ftp/pub/flor2.tex) R. Floreanini, J. LeTourneux and L. Vinet, An algebraic interpretation of the continuous big q-Hermite polynomials. (see /ftp/pub/flor3.tex) Topic #23 ----------------- OP-SF NET ------------------ May 17, 1995 From: OP-SF Net editor <thk@fwi.uva.nl> Subject: Obtaining back issues of OP-SF Net Back issues of OP-SF Net can be obtained from ftp: ftp.fwi.uva.nl, in directory pub/mathematics/reports/Analysis/koornwinder/opsfnet.dir or WWW: http://math.ohio-state.edu/JAT Contributions to the OP-SF Net 2.4 should reach the email address poly@siam.org before July 1, 1995. In order to join the SIAM Activity Group on Orthogonal Polynomials and Special Functions, and thereby receive the Newsletter, you have to become a member of SIAM. Contact the email address service@siam.org. Topic #24 ----------------- OP-SF NET ------------------ May 17, 1995 From: Renato Spigler <spigler@s2.ima.umn.edu> Subject: Abstract of Spigler's talk given at San Diego minisymposium Most of the abstracts of the minisymposium "Special functions and asymptotics" at the 1994 SIAM Annual Meeting during July 25-29 in San Diego were included in the Newsletter of our Activity Group (Fall 1994 and Winter 1994 issues). Here follows a belated abstract of the lecture by Renato Spigler. "Discrete Liouville-Green Approximations and Orthogonal Polynomials Asymptotics" by Renato Spigler(1) and Marco Vianello(2) Abstract of the talk given by R.Spigler at the SIAM Annual Meeting in San Diego,CA,July 1994. Discrete versions of the Liouville-Green (or WKB) asymptotic approximation theorems,first proved by F.W.J.Olver,have been established in certain classes of linear second-order difference equations. These are given by cap delta-squared y_n + q_n y_n = 0, with q_n = a + g_n,a being real positive,zero,or negative (but different from),and g_n a sequence "asymptotically small",in some sense. The difference equation above was shown to represent a kind of canonical form for three-term linear recurrences ( apart from some exception). Applications to the special three-term linear recurrences defining families of orthogonal polynomials in the so-called Blumenthal class,have been made in order to obtain the asymptotic behavior of the polynomials n goes to infinity, with x a fixed parameter. Precise estimates for the error terms accompany these approximations as in the celebrated WKB theorems proved by Olver in the continuous case. References See R.Spigler and M.Vianello,SIAM J. Math. Anal., March 1994 issue (dedicated to R.Askey and F.W.J.Olver) and references therein. (1) Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universita' di Padova,Via Belzoni 7,35131 padova (Italy). e-mail: spigler@ipdudmsa.bitnet , na.spigler@na_net.ornl.gov (2) Dipartimento di Matematica Pura e Applicata,Universita' di Padova,Via Belzoni 7,35131 Padova (Italy). e-mail: marcov@euler.math.pdmat1.it Topic #25 ----------------- OP-SF NET ------------------ May 17, 1995 From: Richard Askey <askey@math.wisc.edu> Subject: Gabor Szego - One hundred years (long topic) A memorial celebration of the 100-th anniversary of Gabor Szego's birth (on January 20, 1895 in Kunhegyes, Hungary) was held on January 21, 1995 in Kunhegyes. See OP-SF Net 2.1, Topic 6 for a short account. One of the speeches delivered there was an essay written by Dick Askey. It was translated into Hungarian and it was read at the celebration by Gyuri Petruska. The essay follows below (in English), with all diacritical (TeX) signs suppressed. GABOR SZEGO - ONE HUNDRED YEARS by Richard Askey It is 100 years since Gabor Szego was born and 80 years since the publication of his first paper. Before this publication he had already shown a strong talent in mathematics by winning the Eotvos competition in 1912. His first paper contained the solution of a problem of George Polya. However, he had earlier, in 1913, published the solution of another problem of Polya. For the nonmathematicians it should be remarked that there are problems at various levels. Some, like those you did in school, are ones which everyone should learn how to do. Then there are contest problems, like those in the Eotvos competition. These are harder and frequently require deeper insight than seems so at first reading. The problem of Polya which Szego solved and published in 1913 is an example of a harder type, which has long attracted prospective mathematicians. Hungary has long specialized in the use of problems to attract young students to mathematics, and other countries have learned from you and have contests of mathematics problems to encourage students to think harder than is needed for the typical school problem. The problem of Polya which made up Szego's first published paper was an open problem and there is still great interest in extensions of Szego's solution to more complicated problems of this nature. Since Szego had been a mathematical prodigy himself, he was an ideal person to be asked to tutor one of the great mathematical minds of this century, John von Neumann. Here is what Norman Macrae wrote in his book "John von Neumann". "Professor Joseph Kurschak [of the Lutheran School in Budapest] soon wrote to a university tutor, Gabriel Szego, saying that the Lutheran School had a young boy of quite extraordinary talent. Would Szego, as was the Hungarian tradition with infant prodigies, give some university teaching to the lad? "Szego's own account of what happened was modest. He wrote that he went to the von Neumann house once or twice a week, had tea, discussed set theory, the theory of measurement, and some other subjects with Jancsi, and set him some problems. Other accounts in Budapest were more dramatic. Mrs. Szego recalled that her husband came home with tears in his eyes from his first encounter with the young prodigy. The brilliant solutions to the problems posed by Szego, written by Johnny on the stationery of his father's bank, can still be seen in the von Neumann archives in Budapest." Macrae was wrong in saying Kurschak was at the Lutheran School. He was a professor at the university. He was the appropriate contact between the mathematics teacher at the Lutheran School and Szego. Szego served in the army in the First World War, but continued to do mathematics, and received his Ph.D. in 1918 in Vienna. Fifty years later he returned to Vienna for a celebration of this, and I still remember how pleased he was with this when he described it to me a few years later. After temporary positions in Hungary, Szego went to Berlin in 1921. Polya was is Zurich and they started to work on a problem book. It turned out to contain too much material for one volume, so was published in two volumes. Polya wrote the following about their work together on these books. "It was a wonderful time; we worked with enthusiasm and concentration. We had similar backgrounds. We were both influenced, like all other young Hungarian mathematicians of that time, by Leopold Fejer. We were both readers of the same well directed Hungarian Mathematical Journal for high school students that stressed problem solving. We were interested in the same kind of questions, in the same topics; but one of us knew more about one topic and the other more about some other topic. It was a fine collaboration. The book `Aufgaben und Lehrsatze aus der Analysis, I, II', the result of our cooperation, is my best work and also the best work of Gabor Szego." It is hard to argue with Polya's assessment about the quality of the problem books. They set a standard for others who would later write books of problems and no one has come close to the level achieved by Polya and Szego. Not only are their problems interesting and important, they build on each other, so that working the problems in a section allows the reader to grow and learn new mathematics and techniques. Szego spent more than ten years in Germany, first in Berlin as privatdocent and then in Konigsberg as professor. His first two Ph.D. students were in Konigsberg. He was a beloved teacher, and when the situation in Germany became hard and then impossible for Jewish mathematicians, and Jews in general, Szego was one of the last to suffer because he was so highly respected by students and colleagues. While in Berlin, he was awarded the Jules Konig prize in 1924. F. Riesz gave the report for the prize committee, and this is reprinted in Riesz's "Oeuvres completes". In 1934, Gabor Szego moved to the United States, first to St. Louis, Missouri, where he taught for four years at Washington University, and then to Stanford, California, where he was chairman of the mathematics department for 15 years, building it into an excellent department. At Washington University, Szego wrote the other great book of his, the first and still the best book on orthogonal polynomials. The study of these polynomials started in the 19th century, and continues to the present. In the 1920's, Szego found a variant on the earlier work, and one of facts he discovered was eventually used in speech synthesisers. A Ph.D. student from Stanford, Paul Rosenbloom, wrote about life as a student under Polya and Szego. In addition to the mathematical education he received, Szego looked out for his cultural education, giving Rosenbloom a ticket to Bartok's concert at Stanford. In 1952, Szego published an extension of his first paper. About this paper Barry McCoy wrote: "It is easily arguable that, of all Szego's papers, `On Certain Hermitian Forms Associated with the Fourier Series of a Positive Function' has had the most applications outside of mathematics. In the first place, the problem which inspired the theorem was propounded by a chemist working on magnetism. Extensions of this work made by physicists have led to surprising connections with integrable systems of nonlinear partial difference and differential equations. In addition Szego's theorem has recently been used by physicists investigating quantum field theory." One way mathematicians are honored is to have something they discovered named after them. There is now the Szego kernel function, the Szego limit theorem and the strong Szego limit theorem, Szego polynomials orthogonal on the unit circle, the Szego class of polynomials. Another way we show that the work of a mathematician is deep enough to last is to publish their selected or collected works. Szego's "Collected Works" were published in 1982. I first met Gabor Szego in the 1950's when he returned to St. Louis to visit old friends, and I was an instructor at Washington University. Earlier, when I was an undergraduate there, I had used a result found by Hsu in his Ph.D. thesis at Washington University under Szego. This was in the first paper I wrote. While at the Univ. of Chicago in the early 1960's, Szego visited. I still remember seeing him at one end of the hall and a graduate student, Stephen Vagi, at the other end of the same hall. They walked toward each other and both started to speak in Hungarian. I am certain they had not met before, and I have always wondered how Szego recognized another former Hungarian. In 1972 I spent a month in Budapest and Szego was there. We talked most days, and though his health was poor and his memory was not as good as it had been a few years earlier, we had some very useful discussions. Three years earlier, also in Budapest, Szego had mentioned two papers of his which he said should be studied. I did not do it immediately, but three months later did. One contained the solution of a problem I had been trying to solve for three years. His paper had been written 40 years earlier. I learned from this that when a great mathematician tells you to look at a paper of theirs which they think has been unjustly neglected, one should do it rapidly. Szego left a memorial for us, his mathematical work. It continues to live and lead to new work. One of his main areas was orthogonal polynomials and this is now a very active field. I often regret that he is not here to appreciate all of the work being done on problems he started. Topic #26 ----------------- OP-SF NET ------------------ May 17, 1995 From: Nico M. Temme <nicot@cwi.nl> Subject: Remarks on Problem 7, The incomplete Airy function (latex file) Editor's note: Below follows a latex file with Remarks by Nico Temme on Problem 7 (The incomplete Airy integral), which was submitted to the Newsletter by E.D. Constantinides and R.J. Marhefka, and which was last printed there in the Winter 1994 issue. ----------------------- beginning of latex file ----------------------- \documentstyle[twocolumn]{article} \topmargin -0.5in \oddsidemargin -0.375in \evensidemargin -0.375in \textheight 9.2in \textwidth 7.25in \columnsep 0.25in \parskip 0.02in \begin{document} \title{Remarks on Problem 7, The incomplete Airy function} \author{Nico M. Temme, CWI, Amsterdam, The Netherlands} \date{\today} \maketitle % % Text goes here * * * % Let $$% I_0(\sigma,\gamma;k):=\int_\gamma^\infty e^{ik(\sigma z+z^3/3)}\,dz $$%% where $k, \sigma,\gamma $ are real, $\sigma<0$. The proposers ask for an asymptotic expansion for large values of $k$ that holds uniformly for $\gamma$ in the neighborhood of the saddle points at $\pm\sqrt{-\sigma}$. The saddle points are widely separated because $\sigma $ is bounded away from 0. It is not possible to construct an expansion as simple as shown in the problem: $$% I_0(\sigma,\gamma;k)\sim \sum_{n=0}^\infty k^{-n}f(\sigma,\gamma,n), $$%% in which $f(\sigma,\gamma,n)$ is expressed in terms of known and easily computed functions. To give a proper description of the asymptotics a Fresnel type integral should be used. Standard methods to obtain such expansions are available, and I will give only the main steps for deriving the expansion. \par First, assume that $\gamma\ge0$. When $\gamma<0$ it is better to use the complete integral (indeed, an Airy function, which is exponentially small as $k\to\infty$) plus $I_0(\sigma,-\gamma;-k)$; the latter can be treated in the same way as the original $I_0$.\par Second, transform the phase function into a simpler one: write $z=w\sqrt{-\sigma}$ to obtain $$% I_0(\sigma,\gamma;k)=\sqrt{-\sigma}\int_\delta^\infty e^{i\kappa( -w+w^3/3)}\,dw $$%% where $\delta =\gamma/\sqrt{-\sigma},\ \kappa= k\,(-\sigma)^{3/2}$. We assume that $\kappa $ is large (we see that $\sigma$ may be small, in some sense).\par Third, transform the phase function into a quadratic function by writing $-w+w^3/3=t^2-2/3$ with $\hbox{\rm sign}(t)=\hbox{\rm sign}(w-1)$. This maps the saddle point at $w=1$ to the saddle point at $t=0$ and gives a standard form $$% I_0(\sigma,\gamma;k)=\sqrt{-\sigma}\ e^{-2i\kappa/3} \int_\varepsilon^\infty e^{i\kappa t^2} f(t)\,dt, $$%% where $$% \varepsilon^2 =2/3-\delta+\delta^3/3, \quad \hbox{\rm sign}(\varepsilon)=\hbox{\rm sign}(\delta-1), $$% and $f(t)=dw/dt=2t/(w^2-1)$. The assumption $\gamma\ge0$ implies $\varepsilon\ge-\sqrt{2/3}$. The function $f$ is singular at $t=-2/\sqrt{3}$ (this point corresponds to $w=-1$, the other saddle point.\par Fourth, use integration by parts. The first step is to write $f(t)=[f(t)-f(0)]+f(0)$. We have $f(0)=1$, and this term yields the Fresnel integral $\int_\varepsilon^\infty e^{i\kappa t^2}\,dt$. The other term can be used for integrating by parts: $$ \int_\varepsilon^\infty e^{i\kappa t^2} [f(t)-f(0)]\,dt={1\over 2i\kappa}\ \int_\varepsilon^\infty {f(t)-f(0)\over t}\,d e^{i\kappa t^2}. $$ The integrated term vanishes at infinity, because $f(t)={\cal O}(t^{1/3})$ as $t\to\infty$. The procedure can be repeated and this gives eventually % % \begin{eqnarray*} &&I_0(\sigma,\gamma;k)\ \sim\ \sqrt{-\sigma}\ e^{-2i\kappa/3} \times\\ && \left[\int_\varepsilon^\infty e^{i\kappa t^2}\,dt\ \sum_{n=0}^\infty{A_n\over (2i\kappa)^n} +{e^{i\kappa\varepsilon^2}\over 2i\kappa} \sum_{n=0}^\infty{B_n(\varepsilon) \over (2i\kappa)^n}\right], \end{eqnarray*} % as $\kappa\to\infty$, uniformly with respect to $\varepsilon\in[-\sqrt{2/3},\infty)$, with $$A_n=f_n(0),\ B_n(\varepsilon)={f_n(\varepsilon)-f_n(0)\over \varepsilon}$$ and $$f_{n+1}(t)=-{d\over dt}{f_n(t)-f_n(0)\over t},\ n=0,1,2,\ldots,\ f_0(t)=f(t).$$ A similar procedure for non-oscillating integrals is given in my paper in SIAM J. Math. Anal. {\bf 13}, No. 2, 239--253, 1982. \vfill \end{document} ----------------------- end of latex file ----------------------- 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 - - Send address changes to: poly-request@siam.org - - Get back issues by ftp from: ftp.fwi.uva.nl, in directory - - pub/mathematics/reports/Analysis/koornwinder/opsfnet.dir - - Information on joining SIAM - - and this activity group: service@siam.org - o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o - The Officers of the Activity Group are: - - - - Charles Dunkl, Chair - - George Gasper, Vice Chair - - Tom H. Koornwinder, Secretary and OP-SF Net Editor - - Martin E. Muldoon, Program Director - o - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - o

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