Volume 2, Number 3

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-                                                                   -
-                                           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      -
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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:

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

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:


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

Francisco Marcellan (EPS Carlos III, Madrid, Spain)
Andre Ronveaux (Facultes Universitaires N.D. de la Paix, Namur,

Date: Tuesday, July 4, 1995.  Time: 15:30


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

               Martin E. Muldoon
               (Program Director, SIAM Activity Group on Orthogonal
               Polynomials and Special Functions)
               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.


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

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
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,

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

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):


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.


   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


or via the AMS e-MATH home page


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

Readers can search the BibNet database or download individual author
bibliographies by directing their favorite Web browsers to


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
  c) By ftp to
     anonymous login, with password your address. Do cd lists/at-net
     to get into the correct subdirectory, and then the usual downloading
  d) All previous Bulletins are presently archived at the home page of 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:
   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.


AMS e-MATH homepage:
WWW:  http://e-math.ams.org

(*) AMS preprint server:
WWW: http://e-math.ams.org/web/preprints/

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:

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

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


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

(*) 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
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
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


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
     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
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

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


                 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

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.


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
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

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

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
\topmargin -0.5in
\oddsidemargin -0.375in
\evensidemargin -0.375in
\textheight 9.2in
\textwidth 7.25in
\columnsep 0.25in
\parskip 0.02in
\title{Remarks on Problem 7, The incomplete Airy function}
\author{Nico M. Temme, CWI, Amsterdam, The Netherlands}
% Text goes here * * *
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
$I_0$.\par Second, transform the phase function into a simpler one: write
$z=w\sqrt{-\sigma}$ to obtain
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,
\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
&&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
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.
end of latex file

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-              Martin E. Muldoon, Program Director                    -
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