Special Functions in the Digital Age

July 22 - August 2, 2002

Material from Talks

George E. Andrews (Department of Mathematics, Pennsylvania State University)  andrews@math.psu.edu

What is needed in Computer Algebra Packages for Mathematical Research!

This is to be the introductory talk for the session on computer algebra. As such it will be a commentary on a few computer algebra packages and what aspects of them seem most helpful to the research mathematician. The talk will touch upon some of the packages that will be introduced and described in subsequent presentations and talks. The philosophy to be presented is that the role of computer algebra packages is FIRST, an aid to discovery and only secondarily is it to prove theorems (even though the latter maybe play an essential part in the former).

In order to make this view concrete, I will provide examples of how computer algebra packages can either impede or expedite discoveries. The first examples chosen will concern the Omega package (developed jointly with Paule and Riese and to be demonstrated later in the workshop). First it is briefly explained how the package rapidly produces generating functions for rather complicated partitions. Next we look at the generating function for partitions that form the sides of non-degenerate k-gons. If you let the Omega package do too much, you fail to grasp what is happening. This is based on Bull. Austral. Math. Soc., 64(2001), 321-329.

Our next example, based on a recent Putnam problem, illustrates how the flexibility of computer algebra packages assists research. Here I think we should hold as an ideal Hardy's description of Ramanujan: "But with his memory, his patience, and his power of calculation, he combined a power of generalisation, a feeling for form, and a capacity for rapid modification of his hypotheses, that were often really startling, and made him, in his own peculiar field, without rival in his day." While most of us are vastly inferior to Ramanujan in each of the categories named, we do have the advantage of these wonderful machines that can assist in emulating a few of Ramanujan's qualities. This portion of the talk will be based on Contemp. Math., 291(2001), 11-27.

The talk will conclude with further eclectic samples of the significant interaction of computer algebra with topics that Ramanujan might have found interesting. In this portion we hope to consider some of the marvelous computer algebra summation packages.

Richard Askey (Department of Mathematics, University of Wisconsin-Madison)  askey@math.wisc.edu

Introduction: The Role of Handbooks of Special Functions (Monday, July 22 at 9:00 am)

Some comments will be made about previous handbooks and the impact they have had, as seen by a user of them for almost 50 years.

Richard Askey (Department of Mathematics, University of Wisconsin-Madison)  askey@math.wisc.edu

Orthogonal Polynomials in One Variable (Friday, July 26 at 9:00 am)    Slides

The classical orthogonal polynomials in one variable are fairly well understood. However, when these are extended to more than one variable, no one knows how many different ways this can be done, and how explicitly the polynomials can be found. There will be three talks on orthogonal polynomials. The first will deal with the way some of the classical polynomials arise in quantum angular momentum and its q-analogue. The 3j symbols can be transformed into two different sets of orthogonal polynomials, the 6j symbols into a set of orthogonal polynomials. In all of these cases there are only finitely many polynomials and the natural orthogonality relation is a discrete sum. Each of these sets of polynomials becomes a set of orthogonal polynomials with respect to an absolutely continuous measure when the parameters are changed. It has been shown that the 9j symbols are orthogonal polynomials in two variables, and the weight function has recently been discovered. However, there still is no explicit representation which can be used to make the polynomial character obvious. It is likely that the obvious analytic continuation of the discrete orthogonality will give the measure when parameters have been changed, but this has not yet been shown.

Richard Askey (Department of Mathematics, University of Wisconsin-Madison)  askey@math.wisc.edu

Assessment of DLMF (Digital Library of Mathematical Functions) (Friday, August 2 at 9:00 am)    Slides

The Digital Library of Mathematical Functions will include some but far from all of the known results on special functions which will be used in the future. The editors and authors are making guesses about what will be used in science and engineering. In the past, and almost surely in the future, such guesses have been far to conservative about what will be needed in some scientific fields. Some guesses will be given. Certain sets of orthogonal polynomials in one and several variables and elliptic hypergeometric functions are two examples. Some wild speculations about such things as an integral representation of the double gamma function will be mentioned.

Partitions with gap conditions: some old and new results

I start this talk by reviewing the Euler as well as Rogers-Ramanujan partition theorems. Next, I will give a unified treatment of the Schur and Goellnitz partition theorems using a method of colored integers. Then, I will discuss a recent four-parameter generalization of Goellnitz partition theorem due to Alladi, Andrews, Berkovich. I will describe an essential role played by "Maple" in discovering and proving this new theorem.

Michael Berry (Department of Physics, Bristol University)

Architecture of diffraction catastrophes Slides: pdf    ppt

A hierarchy of functions can be defined by oscillatory integrals constructed from the polynomial normal forms of catastrophe theory, each with a different topology of coalescence of saddle-points as parameters vary. Diffraction catastrophes fall outside the more familiar hypergeometric class. They have many applications throughout wave physics, where they describe waves (sound, light, water, quantum) near the geometrically stable caustic singularities of ray physics; the description is asymptotic, and gets better as the wavelength gets smaller. Diffraction catastrophes have interesting and beautiful mathematical properties: scaling laws, nonlinear integral identitites, bifurcation and Stokes' sets, geometry of maxima, phase singularities, and as powerful building-blocks of uniform asymptotic expansions. Their importance was not appreciated when the NBS Handbook of Mathematical Functions (Abramowitz and Stegun) was written in the 1960s. Now they will feature in a chapter being written with Christopher Howls for the new NIST Digital Library of Mathematical Functions project.

Ronald F. Boisvert (Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD, USA)  boisvert@nist.gov

Building the DLMF: Information Technology Issues    Slides: html pdf

The Digital Library of Mathematical Functions is envisioned as a versatile Web-based resource of information on the special functions of applied mathematics. The primary content will be carefully researched and verified formulas and graphs providing quick access to the detailed properties of these functions most useful in practical application in science and engineering. Presenting such information in a way that is not only natural and convenient in a Web environment, but also provides capabilities that exceed what is available using traditional publication methods, remains a technical challenge.

Some of the technical issues that must be addressed in such an undertaking include the following:

(a) enabling convenient on-line browsing,
(b) on-line indexing,
(c) search in mathematical databases,
(d) layering of information,
(e) cut-and-paste of formulas,
(f) interactive graphics,
(g) table generation,
(h) alternate views,
(i) application modules,
(j) representation of mathematical formulas on the Web,
(k) interaction with users,
(l) tracking changes,
(m) continued maintenance.

A survey of these challenges, and how they are being addressed in the context of the DLMF will be the main subject of this presentation. Several will be discussed in much greater detail by other speakers at the workshop.

One unique tension between the desire to provide versatile interactive Web content and the requirement to present certified standard reference information arises in the context of graphics. I will illustrate some of the pitfalls using a Java applet for exploring functions of a single variable, and suggest techniques of "honest plotting" which are necessary for their resolution. This will also expose new needs for algorithms and software for special functions.

Finally, I will provide a demonstration of the current working DLMF Web site.

David M. Bressoud (Mathematics and Computer Science Department, Macalester College)  bressoud@macalester.edu  http://www.macalester.edu/~bressoud

Where does Combinatorial Analysis fit in?    Slides:   pdf    ps

This talk will look at specific examples of the interplay between special functions and combinatorial analysis. The intention is to engender discussion of the role of a chapter on Combinatorial Analysis within the Digital Library of Mathematical Functions.

Bruno Buchberger (Research Institute for Symbolic Computation (RISC), Johannes Kepler University, A4040 Linz, Austria)  buchberger@risc.uni-linz.ac.at

Theorema and Mathematical Knowledge Management    Slides:    pdf    ps

Identities for special functions are just special formulae in predicate logic whose correctness has been established, typically, by proofs produced by human mathematicians. Recent advances in various areas of mathematics have made it possible to invent and/or prove many of these identities by algorithms. In this talk, we put these advances into the more general context of  formal, computer-supported mathematics, notably automated theorem proving. We give an overview on the Theorema system, which aims at providing algorithms for proving, solving, and simplifying classes of formulae  in various areas of mathematics in a uniform logic and software technologic frame. We describe future mathematics as the process of "mathematical knowledge management" in wich mathematical theories on a meta-level establish individual theorems that form the basis of proving, solving, and/or simplifying infinite collections of formulae on a lower level. We propose to build up mathematical knowledge bases for all areas of mathematics and to provide tools for the formal manipulation of these knowledge bases that are essentially based on automated proving.

We will demonstrate the design and current capabilities  of Theorema by a couple of examples. Theorema is programmed in Mathematica and, thus, is available on all platforms.

Peter A. Clarkson (Institute of Mathematics & Statistics, University of Kent, Canterbury, CT2 7NF, UK)  P.A.Clarkson@ukc.ac.uk

The Painleve equations - nonlinear special functions     Expanded version of the talk and figures: SF02L.pdf    SF02L.ps    plots.pdf    plots.ps

The six Painleve equations (PI-PVI) were first derived around the beginning of the twentieth century by investigation by Painleve, Gambier and their colleagues in a study of nonlinear second-order ordinary differential equations. There has been considerable interest in Painleve equations over the last few years primarily due to the fact that they arise as reductions of soliton equations solvable by inverse scattering. Further, the Painleve equations are regarded as completely integrable equations and possess solutions which can be expressed in terms of the solutions linear integral equations. Although first discovered from strictly mathematical considerations, the Painleve equations have appeared in various of several important applications including statistical mechanics, random matrices, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics. The Painleve equations may also be thought of as nonlinear analogues of the classical special functions such as Bessel functions. Their general solutions are transcendental in the sense that they cannot be expressed in terms of previously known functions. However, for special values of the parameters, PII-PVI possess rational solutions and solutions expressible in terms of special functions. For example, there exist special solutions of PII-PVI that are expressed in terms of Airy, Bessel, parabolic cylinder, Whittaker and hypergeometric functions, respectively. Further the Painleve equations admit symmetries under affine Weyl groups which are related to the associated Backlund transformations. In this talk I shall give an overview of some of plethora of remarkable properties which the Painleve equations possess (including connection formulae, Backlund transformations, associated discrete equations and hierarchies of exact solutions) and some of their applications.

Edgardo S. Cheb-Terrab (MITACS - CECM - Simon Fraser University, Canada Symbolic Computation Group, UW, Canada, Theoretical Physics Department, UERJ, Brazil)  ecterrab@cecm.sfu.ca  http://lie.uwaterloo.ca/ecterrab
Maple library for Special Functions http://lie.uwaterloo.ca/pub/mathfuncs

Special functions & Maple

A flexible conversion facility in the mathematical language is as important as a dictionary in spoken languages. Such a tool is implemented in the Maple system as a net of conversion routines aiming at expressing any mathematical function in terms of another one, whenever that is possible as a finite sum of terms. When the parameters of the functions being converted depend on symbols in a rational manner, any assumptions on these symbols - e.g. made with the "assuming" facility - are taken into account at the time of performing the conversions.

Most special functions also arise as solutions to some differential equations - ODEs or PDE - of linear and non-linear type, polynomial in the unknowns and their derivatives. This polynomial differential representation of a mathematical function is the starting point for establishing the function's properties. By composing functions with themselves, arbitrary powers, additions and products one obtains rather arbitrary non-polynomial objects which can also be represented in differential (typically non-linear) polynomial form. The routines being presented can construct these differential polynomial representations in general, making possible the computation of subtle identities, the solving of non-polynomial (possibly non) differential systems using techniques for differential polynomial ones, etc. This tool is at the root of the current Maple PDE system solver.

Finally, the requirement concerning mathematical functions is not merely computational: typically, one needs information on established identities, alternative definitions and mathematical properties in general. We usually look for that information in handbooks like Abramowitz & Stegun. Part of this information is already found in internal Maple subroutines. This motivates the idea of a "function wizard" project, whose main purpose is to provide access to each piece of this information through a simple interface, including the goal of making the information complete. Such a "computer algebra handbook of mathematical functions" - a concept close to that of a live function wizard - is an ongoing project expected to superseede textbooks at some moment, in that it can respond to requests by processing whatever (growing number of) mathematical information using (a growing number of) mathematical algorithms.

Mgfun, a Maple Package for Symbolic Summation and Integration of Special Functions and Combinatorial Sequences    Slides:    minneapolis02.pdf    minneapolis02.ps    sfda.pdf    sfda.ps

A large class of special functions and combinatorial sequences that are implicitly represented by systems of linear functional equations is amenable to computer algebra methods. The class enjoys many closure properties that have recently been turned into algorithms, now implemented in the Maple package Mgfun. The presentation treats concrete examples with our implementation. Applications include the evaluation of parametrized definite integrals and sums, series and asymptotic expansions, and the automatic proof of identities.

Related link:

http://algo.inria.fr/chyzak/mgfun.html

Charles F. Dunkl (Department of Mathematics, University of Virginia)  cfd5z@virginia.edu  http://www.math.virginia.edu/~cfd5z/

Orthogonal Polynomials on the Sphere Paper form of the Slides:   pdf    ps

The Dirichlet problem for the sphere requires the determination of a function harmonic in the interior with specified boundary values. One approach to the solution is to expand the boundary values as a series of spherical harmonics. These are the restrictions of harmonic homogeneous polynomials to the surface. Other partial differential equations with spherical symmetry can be solved by using spherical harmonics, for example the field of a point charge in a hollow sphere, or the wave function of an electron subject to a Coulomb potential. The classical orthogonal polynomials of the Legendre, Gegenbauer and Jacobi families appear in specific formulae for the harmonics. In the 60's and 70's it became apparent that harmonic analysis, that is, the study of functions on which there are group actions, has a natural setting on the sphere. Perhaps the first important result was to realize Gegenbauer polynomials as spherical functions on the sphere as a homogeneous space of the rotation group SO(N). So for a certain period of time the emphasis was on decomposing the space of spherical harmonics of a given degree according to the action of parabolic subgroups like SO( N-1) and SO( m) × SO(N-m). This provided a setting and elegant proofs for product formulae and addition theorems. Even in those times the importance of finite symmetry groups was already manifest. In order to analyze the wave functions of electrons in molecules with a crystal structure it is necessary to use the structure of spherical harmonics invariant under the symmetry point group of the crystal. Later it was realized that finite reflection groups and root systems are more fundamental than the rotation group. Gegenbauer and Jacobi polynomials appear as the spherical harmonics associated to reflection groups of rank 1 or 2. Now we have larger classes of weight functions on the sphere. The classical theory of spherical harmonics can be taken over to weight functions consisting of products of powers of linear functions invariant under a reflection group. In the past we used the idea of irreducible unitary representations to produce orthogonal decompositions, nowadays we construct commutative algebras of self-adjoint operators whose simultaneous eigenfunctions are the orthogonal polynomials we wish to study. This is completely successful for the weight function consisting of powers of the coordinate functions, and gives a nice basis of products of ordinary Jacobi polynomials. It is a more difficult matter to find bases of polynomials with, for example, hyperoctahedral invariance. A novel basis for symmetric functions is introduced to solve this problem; although the obtained basis is not orthogonal we can find the inner products and explicitly compute the determinant of the Gram matrix. The harder problem of spherical harmonics associated to Calogero-Moser problems will be discussed in connection with the by now well-developed theory of symmetric and nonsymmetric Jack polynomials.

Walter Gautschi (Department of Computer Sciences, Purdue University)  gautschi@inf.ethz.ch

Orthogonal Polynomials (in Matlab)    Slides

This will be a survey talk on numerical methods developed in the last 30 years or so for computing orthogonal polynomials and related problems.

Mourad Ismail (Department of Mathematics, University of South Florida)  maismail@ust.hk

Continued Fractions and Biorthogonal Functions    Slides

We discuss T and R fractions and how they naturally lead to orthogonal and biorthogonal rational functions. Plans for future research in this area will be discussed.

Alexander R. Its (Indiana University-Purdue University Indianapolis)  a.its@ic.ac.uk

Integrable Systems and Integrability

The goal of this talk is twofold. The first and the main objective is to present an overview of the modern theory of integrable systems. The theory was originated in the remarkable work of Gardner, Green, Kruskal, and Miura of 1967 on the Korteweg-de Vries equation. Since then it has gradually transformed into a subject which could be called the nonlinear special functions and which overlaps now with many areas that have never been considered before as "integrable systems."

The focus of the talk will be on the analytic aspects of the theory of integrable systems represented by its principal analytic ingredient - the Riemann-Hilbert method. We will argue that the method can be thought of as a non-commutative analog of the method of contour integral representations. Applications of the Riemann-Hilbert technique to the asymptotic analysis of the Painlevé transcendents, including rigorous derivation of the relevant connection formulae, will be discussed in detail.

The second objective of the lecture is, to a certain extent, of a speculative nature. We will try to use the main topic as an opportunity to reflect on the very notion of "integrability." In fact, we shall try to go beyond the classical definitions of integrability in the sense of Liouville and Frobenius. An ideal goal would be a rigorous understanding of such commonly used terms as "explicit solution,'' "exact formula,'' etc. Most certainly we are, at the moment, very far from even a rigorous formulation of the question. Still, some relevant observations toward the goal mentioned can be made, and we will try to do so when discussing the recent applications of the Riemann-Hilbert method in matrix models, special functions and combinatorics.

Tom H. Koornwinder (Kdv Institute for Mathematics, Universiteit van Amsterdam)  thk@science.uva.nl

Algebraic methods: Lie groups, quantum groups    Expanded version:    pdf    ps

In the Bateman project, special functions in connection with group theory only figured in the (excellent) chapter 11 on spherical harmonics. When this project was underway, some fifty years ago, more was already known about special functions arising in group representations. In particular, the pioneering work of Wigner has been important. Since then there has been an increasing interaction between special functions and group representation theory. The profits of this interaction for special funtion theory were a new way of systematizing the theory, new conceptual proofs of old results, new results for known special functions, and the introduction of completely new classes of special functions. On the other hand, in group representation theory and in harmonic analysis on groups (in particular noncompact semisimple Lie groups) certain results essentially used hard analytic facts about special funtions occurring in that context. Applications of the interaction between special functions and group theory were made in particular in physics (for instance Clebsch-Gordan coefficients), while conversely physical motivation often led to new examples of this interaction.

Until the late eighties, the development of this interaction went along two tracts. A tract started by Wigner and vigorously extended by Vilenkin studied matrix elements of group representations as special functions. A special case of this approach focused on spherical functions. This was a source of product formulas and addition formulas, see work by, among others, Koornwinder, D. Stanton, Dunkl. A spin-off was the idea of special functions associated with root systems, which led to the Heckman-Opdam hypergeometric functions, the Dunkl operators and the Macdonald polynomials.

A second tract, having earlier roots but brought to full maturity by W. Miller Jr., started with a family of special functions and differential-difference operators associated wih them. The operators generate a Lie algebra and hence a local Lie group. This again naturally implies many formulas for the special functions one started with, for instance generating functions. In later work, Miller focused on symmetry aspects of separation of variables.

The introduction of quantum groups during the eighties had an enormous impact on the theory of q-special functions which, a little earlier had got a new dynamics by, among others, the discovery of Askey-Wilson polynomials. While q-special functions, until then, only were found to live on p-adic groups and Chevalley groups, they now got a very natural setting on quantum groups. To some extent, everything which had been done earlier for special functions in connection with Lie groups could now be repeated in the q-case, but more exciting phenomena were met because the q-universe is not just a dull parallel of the classical universe, but one object in the q-universe can correspond to different classical objects, and conversely.

During the nineties Macdonald polynomials and Koornwinder's extension of them found their setting in the quantum world in various quite different ways. Much impact had Cherednik's interpretation of Macdonald polynomials in the context of affine Hecke algebras. Representation theory of (affine) Hecke algebras and harmonic analysis on them became important for special function theory.

Important recent developments involve representation theory of noncompact quantum groups, the study of dynamical quantum groups, and work on elliptic quantum groups.

Christian Krattenthaler (Institut fur Mathematik, Universitat Wien)  kratt@ap.univie.ac.at

HYP and HYPQ    Matematica Slides:   hyp.nb    hyp1.nb    hypq.nb
Pdf Slides:   hyp.pdf    hyp1.pdf    hypq.pdf

I shall demonstrate the main features of the Mathematica packages HYP and HYPQ, which are designed for convenient manipulation of binomial and q-binomial sums, and hypergeometric and basic hypergeometric series. In particular, the packages contain an extensive built-in list of summation and transformation formulas for these series.

The packages can be downloaded at

http://www.mat.univie.ac.at/~kratt/hyp_hypq/hyp.html

Daniel Lozier (National Institute of Standards and Technology (NIST))  lozier@nist.gov

Development of a New Handbook and Web Site of Properties of Special Functions    Slides:   html    pdf    powerpoint

Abramowitz and Stegun's Handbook of Mathematical Functions, published in 1964 by the National Bureau of Standards, is familiar to many mathematicians. It is even more familiar to scientists who use special functions in their daily work. But developments in mathematics and computing since 1964 make the old handbook less and less useful. To meet the need for a modern reference, NIST is developing a new handbook that will be published in 2003. A Web site covering much the same territory will be released also. This work is supported by the National Science Foundation. Details of the project and its current status will be described.

Oleg Marichev (Wolfram Research, Inc.)

www.functions.wolfram.com - The web's most comprehensive site about special functions

Joint work with Michael Trott.

At the Mathematical Functions website, we have a collection of most of the known formulas for practically all mathematical functions. It is organized in such a way as to allow for expansion, citation, searching, conversion of formulas, and substantial computer support from the Mathematica system.

Currently (June 2002) the site includes more than 37,300 formulas for the approximately 250 elementary and special functions that are available in(Mathematica. It is already the biggest online collection of formulas presented in different format types ( notebooks, HTML, and PDF). The site also has a well-developed, unified structure; in particular, the organization of mathematical files and programming solutions allows for semiautomatic updating and citation of formulas. The site contains thousands of new or adapted-to Mathematica formulas for elementary and classical special functions.

In this talk we will outline some of the material at the Mathematical Functions site and also discuss how it was put together.

Bruce R. Miller (National Institute of Standards and Technology (NIST))  bruce.miller@nist.gov

Representation, display and manipulation of mathematics on the Web    pdf    ps    Slides:    pdf

Earlier in this workshop, we will have seen how MATHML provides a means to display mathematics on the web. We will have heard about automatic verification of special function identities using computer algebra and theorem proving software. TEX (and LaTEX) sets the standard for quality typesetting of mathematics. Presentation MATHML has the potential to approach its quality, as implementations mature and consistent fonts become available, and this must be a goal. Yet, online mathematics also offers the possibilities of various manipulations of the formula by the user ranging from simple selections of alternative forms of expressions, through broad rearrangements, to extensive computations with and validations of formula. What is required of a representation of mathematics that will support these, and other foreseeable, capabilities?

An ideal representation must be able to capture the nuances of quality display as well as the precise meaning of the formula. The presentation hints include the `exceptions' that an author introduces to make it's meaning more apparent to a reader, such as the ordering of variables and terms, presence and omission of parenthesis in certain situations, choice among different notation for the same operations and so on. While TEX, LATEX and Presentation MATHML easily capture this presentation information, they sidestep the issue of the meaning of formula. Which F is being referred to? What does a given prime indicate? A useful representation must also capture the meaning in sufficient precision that formulas can be supplied as input to computer algebra systems and theorem provers without fear of misinterpretation. Content MATHML and OPENMATH convey the semantics (and the input languages of computer algebra systems each convey their own version of the semantics), but, alone, lack the presentation information.

A third aspect, metadata, must also be considered. Irrespective of how well a mathematical search engine can be made to work, there are facets to a formula that cannot be inferred from the formula itself. For example, the role of a formula as an addition theorem, or asymptotic expansion, may best be handled by annotating the formula, perhaps with simple textual information. Additional information regarding the dependence of one formula on another, or constraints on variables, or the derivation of the formula may be better handled by structured annotations, such as provided by the OMDOC extension of OPENMATH.

Thus one is led to a core representation consisting of a hybrid of presentation and content markup, along with additional annotations for metadata. Since the representation most appropriate for exchange, delivery and presentation in different media may be different, the representation must support the transformation from one the core format to another. The various XML formats (MATHML and OPENMATH) are designed to support this.

But, given the verbosity of XML, these may not be the the best form for authoring. Indeed, writing being a rather personal thing, the styles of different authors must be accommodated. The mathematical community must therefore develop various means of authoring for content, as well as presentation. Some may be based on XML editors with graphical user interface. Others may be based on computer algebra system worksheets.

Given the longstanding preference for LATEX among most of the DLMF project members, we are developing a LATEX based approach. We extend LATEX by defining macros at a higher semantic level, for example for various calculus operations; macros are defined for each special function, and so on. By minimizing the ambiguities inherent in purely presentational markup, an infix parser with only minimal heuristics should be able to transform the formula into the corresponding semantic representations.

Willard Miller (School of Mathematics, University of Minnesota)  miller@ima.umn.edu  http://www.ima.umn.edu/~miller/

The Lie Theory approach to special functions Expanded version:   pdf     ps

I will discuss some of the most important interrelationships between the theory of Lie groups and algebras, and special functions, with a strong emphasis on results obtained in the 50 years after the publication of the Bateman Project. An informal justification for this treatment is that most functions commonly called "special" obey symmetry properties that are best described via group theory (the mathematics of symmetry). In particular, those special functions that arise as explicit solutions of the partial differential equations of mathematical physics, such as via separation of variables, can be characterized in terms of their transformation properties under the Lie symmetry groups and algebras of the differential equations. (The same ideas extend to difference and q-difference equations.) I shall treat, briefly, the following topics:

1. Special functions as matrix elements of Lie group representations. (addition theorems, orthogonality relations)

2. Special functions as basis functions for Lie group representations (generating functions)

3. Special functions as solutions of Laplace-Beltrami eigenvalue problems (with potential) via separation of variables.

4. Special functions as Clebsch-Gordan coefficients for the reduction of tensor products of irreducible group representations (the motivation for Wilson polynomials).

In practice, the first two items involve hypergeometric functions predominantly and are special cases of the third item. The group theoretic basis for variable separation allows treatment of non-hypergeometric functions, such as those of Lame' and Heun. The last item provides an important motivation for the construction of the Askey-Wilson polynomials.

I will conclude with a brief examination of special functions (or functions that deserve to be called "special") that arise when one restricts certain irreducible Lie group representations to a discrete lattice subgroup. The two most important examples are an irreducible representation of the Heisenberg group (and its relation to the windowed Fourier transform, the Weil-Brezin-Zak transform and theta functions), and an irreducible representation of the affine group (and its relation to the continuous and discrete wavelet transforms). I will describe the properties of the Daubechies family of scaling functions, a very modern family of "special" functions arising as solutions of difference equations.

The accompanying notes contain additional details.

Cleve Moler (Chairman and Chief Scientist The MathWorks, Inc.)  moler@mathworks.com

Special Functions in MATLAB What do we have and what are we missing?

Current versions of MATLAB provide a limited number of special functions. Most of the special function library is based on old Fortran codes by Cody and Amos. The connection through the Symbolic Toolbox to Maple provides a richer set of functions, but this is available to only a fraction of MATLAB users. What other functions should we provide in MATLAB itself, and how should we provide them?

Andrew M. Odlyzko (Department of Mathematics and Digital Technology Center, University of Minnesota)  odlyzko@umn.edu  http://www.dtc.umn.edu/~odlyzko

Zeta functions

The Riemann Hypothesis, which predicts that all non-trivial zeros of the Riemann zeta function lie on the critical line, is the most famous unsolved problem in mathematics. Most of the interest in this function has therefore been concentrated on the zeros, and the latest results in this area will be surveyed. However, the zeta function, and more general classes of zeta functions, also have many other interesting properties as special functions of mathematics, and some of the more interesting ones will be discussed.

Andrew M. Odlyzko (Department of Mathematics and Digital Technology Center, University of Minnesota)  odlyzko@umn.edu  http://www.dtc.umn.edu/~odlyzko

The new (and of necessity digital) libraries for mathematics Slides:    pdf    ps

Traditional journals, even those available electronically, are changing slowly. However, there is rapid evolution in scholarly communication. Usage is moving to electronic formats. In some areas, it appears that electronic versions of papers are being read about as often as the printed journal versions. Although there are serious difficulties in comparing figures from different media, the growth rates in usage of electronic scholarly information are sufficiently high that if they continue for a few years, there will be no doubt that print versions will be eclipsed. Further, much of the electronic information that is accessed is outside the formal scholarly publication process. There is also vigorous growth in forms of electronic communication that take advantage of the unique capabilities of the Web, and which simply do not fit into the traditional journal publishing format.

This lecture will present some statistics on usage of print and electronic information. It will also discuss some preliminary evidence about the changing patterns of usage. It appears that much of the online usage comes from new readers (esoteric research papers assigned in undergraduate classes, for example) and often from places that do not have access to print journals. Also, the reactions to even slight barriers to usage suggest that even high quality scholarly papers are not irreplaceable. Readers are faced with a "river of knowledge'' that allows them to select among a multitude of sources, and to find near substitutes when necessary. To stay relevant, scholars, publishers, and librarians will have to make even larger efforts to make their material easily accessible.

Ingram Olkin (Stanford University)  iolkin@stat.Stanford.EDU

Interface Between Statistics and Special Functions

There is a long history of statistical problems that served to motivate research in linear algebra, in numerical methods, and in special functions. We here focus on topics arising fom multivariate analysis: extremal problems, multivariate integrals and distributions, totally positive functions, and eigenvalue problems.

Frank W.J. Olver (Institute for Physical Science and Technology, University of Maryland and Mathematics & Computational Sciences, National Institute of Standards and Technology (NIST))  olver@ipst.umd.edu

Error Bounds; Hyperasymptotics; Uniform Asymptotics    Slides

A survey is made of the current state of the art in asymptotic analysis, especially with regard to the computation and application of special functions.

Topics to be covered are:

1. Construction of rigorous and readily computable error bounds.
   
   
2. Re-expansion of remainder terms in series involving the complementary error function, incomplete gamma functions, and other functions, with a view to increasing attainable accuracy.
   
   
3. Construction of asymptotic expansions that are uniform with respect to one or more parameters, for example, the degrees or orders of the special functions.

Conclusions will be drawn concerning the most useful directions in which future research in each of these areas is likely to proceed.

Peter Paule (RISC, J. Kepler University Linz, Austria)

Symbolic Summation: Algorithms and Missed Opportunities    Slides

The problem of simplifying complicated sum expressions arises not only in special functions but in many mathematical fields. Nevertheless, symbolic algorithms that assist in this task do not have a very long history.

The starting point of symbolic summation with the computer is Gosper's algorithm (1978), a decision procedure for indefinite hypergeometric summation. Despite being a first breakthrough, for a long time its applicability has been considered as quite restricted since most hypergeometric summations arising in practice are definite ones. Zeilberger's `creative telescoping' (1990) dissolved this limitation. Since then symbolic summation has turned into an active subarea of computer algebra on its own.

This talk presents a survey on algorithms and methods along a historical chain of missed opportunities.

The first talk of the computer algebra session by George Andrews focuses on the role computer algebra tools play in discovery. The primary object of this second talk of the session is to introduce to some of the mathematical ideas which make the algorithmic machinery work. Illustrative examples (e.g., hypergeometric sums and multiple-sums, q-series, harmonic number identities) should whet one's appetite to attend the software presentation in the afternoon.

William P. Reinhardt (Department of Chemistry, University of Washington, Seattle)  rein@chem.washington.edu

New and old addition theorems and Landen identities for Jacobian elliptic functions: do these indeed give rise to "novel" solutions for non-linear PDEs?

The algebra of addition formulae for theta and Jacobian elliptic functions is indeed profuse and bewildering in their notations and multivarious forms. Khare and Sukhatme have recently introduced scores of identities for sums of Jacobi functions with arguments, z, augmneted by addition of 2(i-1)K/p, K being the "quarter period" which is a complete elliptic integral of the first kind, and i =3D 1,2,3,...p. They have investigated p running from 2 to 9. Small values of p give well known results, larger values lead to results which seem not to have been noted before. They have subsequently found expansions for single Jacobi functions in terms of sums of repetitions of the same function with differeing moduli (Landen type identities), also with agruments spaced along the same series z+ 2(i-1)K/p.

What are we to make of these new identities, which were discovered by Khare and Sukhatme (Ref. 1 below) in "finding new solution classes" for non-linear equations, such as the KDV equation? A first clue is that these "new" solutions of the KDV equation have been now found, by the present author, to actually be well expressed in terms of old and well known solutions through yet another apparently novel set of Landen type identities, this time involving relations between squares of the Jacobian functions.

We will attempt to draw all of this together in a manner suitable for electronic presentation: i.e. what is real, novel, and of permanent value, and which parts of this can be systematized in a useful manner? We also attempt to draw out their connections to work of the ancients, namely what is their relationship to the cubic and quartic curve invariants of Abel and Weierstrass, which provide geometric underpinnings to the original addition theorems?

Ref 1: A. Khare and U. Sukhatme, Linear Superpostion in Nonlinear Equations, Phys. Rev. Letts. 88, 244101 (June 17, 2002).

Donald Richards (Department of Statistics, University of Virginia)  dsr2j@unix.mail.virginia.edu

Computers and special functions in multivariate statistical analysis

In this talk, which is based on joint work with Richard McFarland, we shall study two classical problems in multivariate statistical analysis of discriminant analysis and of missing data analysis. In each of these problems it is an old and difficult question to evaluate the exact probability distribution of certain maximum likelihood estimators under the assumption of fixed sample sizes. Using the theory of Bessel and confluent hypergeometric functions of matrix argument, we first show that the probability distributions of these maximum likelihood estimators can be described in terms of algebraic functions of classical random variables. Using these algebraic functions we then apply direct Monte Carlo simulation to evaluate the probability distributions of the maximum likelihood estimators.

Axel Riese (Research Institute for Symbolic Computation, J. Kepler University Linz)  Axel.Riese@risc.uni-linz.ac.at  http://www.risc.uni-linz.ac.at/people/ariese/home/

Computer Proofs of Hypergeometric Summation Identities and Partition Analysis

We demonstrate Mathematica packages related to special functions. The first part deals with hypergeometric summation. We show how Zeilberger's algorithm can be used to automatically prove single summation identities. Furthermore, we present an algorithm which generalizes and improves Sister Celine's technique for proving multiple summation identities. For both algorithms we have worked out q-analogues.

The second part is devoted to MacMahon's Partition Analysis, a method for solving problems related to linear diophantine equations and inequalities. Together with George Andrews, who has algorithmized this method, and Peter Paule we have developed the Omega package, which will be demonstrated by means of several examples.

Our packages are freely available for non-commercial users; see http://www.risc.uni-linz.ac.at/research/combinat/risc/software/

Bonita V. Saunders (National Institute of Standards and Technology, 100 Bureau Drive Stop 8910, Gaithersburg, MD 20899-8910)  bonita.saunders@nist.gov  http://math.nist.gov/~BSaunders

Interactive 3D Visualizations of High Level Functions in a Mathematical Digital Library

A Web-based digital library offers an ideal medium for informative graphical representations of high level mathematical functions. In contrast to 2D and 3D still images in printed media, Web capabilities make it possible to create dynamic interactive visualizations that can help users gain a deeper understanding of special functions.

Still, the development of effective graphics for a digital library presents several challenges. There must be a reliable means for computing the data. A sophisticated and intimate knowledge of the function may be required to determine which features should be emphasized. Furthermore, singularities, poles, branch cuts and other complexities can make the creation of accurate visualizations quite difficult.

This talk will discuss steps taken to address these issues, including the use of numerical grid generation to design suitable computational meshes to plot complicated functions. The visualizations developed for the NIST Digital Library of Mathematical Functions (DLMF) will be compared to graphical presentations in the original NBS handbook and earlier references such as Jahnke and Emde. Also, we will compare the design of graphics for the NIST DLMF with what is available in popular computer algebra systems and suggest some areas where future research is needed.

Otmar Scherzer (Department of Computer Science, University Innsbruck, Techniker Str. 25, A-6020 Innsbruck, Austria)  Otmar.Scherzer@uibk.ac.at

Case examples of Special Functions in Analysis and Numerics

Special functions play a significant role in modern numerical methods. In particular wavelet functions became a key technology in scientific computing for solving partial differential equations via a finite element ansatz or boundary integral methods. These methods can be implemented efficiently using multi-scale properties of wavelets. Moreover, wavelets can be designed with computer algebra methods to optimally approximate the desired solution with a small number of wavelet ansatz functions. Similar ideas are utilized to construct efficient methods for the numerical solution of ordinary differential equations.

It is a fact that summation plays an important role in proving existence of solutions of nonlinear partial differential equations, in particular nonlinear semi-group theory. We summarize the role of special function techniques, like summation and generating functions, from the fundamental paper of Crandall and Liggett for proving existence of nonlinear parabolic partial differential equations.

Finally we present two examples, a variational problem and an inverse problem, where we used spherical harmonics to reduce the originally higher dimensional problems to one dimensional problems. We were experimenting with efficient finite element software to solve the higher dimensional variational problem. With this software we were able to find a qualitatively correct solution - however quantitatively it was wrong. For symmetric test cases, using spherical harmonics we were able to calculate the solution analytically and were able to rescale the higher dimensional problem in order to get a qualitatively correct solution with the finite element code.

Carsten Schneider (Research Institute for Symbolic Computation, J. Kepler University Linz )  Carsten.Schneider@risc.uni-linz.ac.at  http://www.risc.uni-linz.ac.at/people/cschneid/

Sigma: A Summation Package for Discovering and Proving    Slides:    pdf    ps

Sigma is a summation package, implemented in the computer algebra system Mathematica, that enables to discover and prove nested multisum identities. Based on Karr's difference field theory (1981) this package allows to find all solutions of parameterized linear difference equations in a very general difference field setting, so called PiSigma-fields. With this difference field machinery nested indefinite multisums can be simplified by minimizing the depth of nested sum-quantifiers. In addition, Sigma provides several algorithms in order to discover closed form evaluations of definite nested multisums. Here one first tries to compute a recurrence for a given definite sum by applying Zeilberger's creative telescoping idea in the difference field setting. Second one attempts to solve this recurrence in terms of d'Alembertian solutions, a subclass of Liouvillian solutions. Combining these solutions one finally may find a closed form evaluation of a definite multisum. All these aspects will be illustrated by various examples.

Besides discovering multisum identities, Sigma is a very powerful tool to prove definite nested multisum identities. In this software presentation I will illustrate how the user can be dispensed from these proving aspects by embedding Sigma as an "external prover'' in the Theorema-system. Theorema, designed by B. Buchberger, supports various facets of mathematical proving, solving and computing in a human readable fashion. One of our goals is to combine these tools into a complete identity prover.

V.P. Spiridonov (JINR, Dubna and MPIM, Bonn)

Elliptic Hypergeometric Functions    Slides

This is a brief qualitative review of the historical origins, general structure, existing applications, and possible future developments of the theory of hypergeometric type series built out of the Jacobi theta functions.

Topics to be mentioned. Connections between Barnes multiple gamma, q-gamma, and elliptic gamma functions. General structure of the plain and basic hypergeometric functions via ratios of sequential series coefficients and its Jacobi theta functions generalization. Elliptic functions origins of the notions of balancing, well-poisedness, and very-well-poisedness for hypergeometric type series. Modular group invariance. Contiguous relations for a terminating very-well-poised ₁₂E₁₁ elliptic hypergeometric series. Frenkel-Turaev identities (elliptic extensions of the terminating Jackson _{8 7} sum and Bailey _{10 9} transformation). The elliptic beta integral (an elliptic extension of the Askey-Wilson and Nassrallah-Rahman integrals). Elliptic analogues of the Wilson (discrete) and Rahman (continuous) families of biorthogonal rational functions. A non-rational functions generalization and two-index biorthogonality. An elliptic extension of the Ramanujan-Watson-Gupta-Masson continued fraction. Multivariable special functions (elliptic Selberg integrals and some summation formulae). Work in progress and some further possible developments.

(A part of the author's results to be presented was obtained in collaboration with J.F. van Diejen and A.S. Zhedanov.)

Dennis Stanton (School of Mathematics, University of Minnesota)  stanton@math.umn.edu

Exponential formulas     Slides

The exponential formula relates the combinatorial objects for the generating functions f(x) and exp(f(x)). Several settings for exponential formulas exist. In this talk I will review some of them and what is known (and not known) about their q-analogues in permutation enumeration. Some results which should be amenable to computer proofs will be presented.

Nico M. Temme (CWI)  Nico.Temme@cwi.nl    http://www.cwi.nl/~nicot

Numerics of Special Functions    Slides:   pdf

The following topics will be discussed:

• Literature
  
  
• Some simple experiences
  
  
• Numerical aspects


- Insight in function behavior, selecting analytical tools

- Selecting numerical methods, stability of the algorithms

- Efficiency of the algorithms, underflow, overflow, scaling; testing

- High quality software


• Numerical methods
  
  - Power series, asymptotic series, recursions, difference equations
  
  - Chebyshev expansions, continued fractions, differential equations
  
  - Quadrature, Gauss, trapezoidal; uniform asymptotic expansions
  
  - Rational approximations, Pad\'e, Chebyshev sense
  
  - Convergence acceleration
  
  
• Why is new software needed? What has to be done?
  
  
• New approaches
  
  
• Present project: examples of ongoing research (together with Amparo Gil and Javier Segura)

Alexander Turbiner (Instituto de Ciencias Nucleares, National University of Mexico)  turbiner@nuclecu.unam.mx

Algebraic Approaches: Quasi-Exactly-Solvable Problems (Lie-algebraic theory of polynomial solutions of differential and finite-difference linear equations)    abstract.pdf     abstract.ps    Slides: pdf ps

Hans W. Volkmer (Department of Mathematical Sciences, University of Wisconsin)  volkmer@csd.uwm.edu

Mathieu, Lamé and Spheroidal Wave Functions    pdf    ps
Slides

Peter L. Walker (College of Arts & Science, American University of Sharjah)  peterw@aus.ac.ae

The elliptic functions of Jacobi and Weierstrass

The talk will describe the Jacobian and Weierstrassian families of elliptic functions. The emphasis will be on points of view which have recently become important, new results which have come to light as a result of the DLMF project, and graphical representations.

Abdou Youssef (Department of Computer Science, The George Washington University, Washington, DC 20052)  youssef@seas.gwu.edu

Search Systems for Mathematical Equations    pdf    ps

To process and disseminate technical knowledge more effectively, efforts are underway worldwide to create and codify Web-accessible digital libraries of mathematical contents. Notable examples include the Digital Library of Mathematical Functions (DLMF) project at the National Institute of Standards and Technology (NIST), and the markup languages MathML and OpenMath. To benefit from such digital libraries, users should to be able to search not only for text, but also for equations and other math constructs.

Searching can be divided into three broad classes of increasing complexity: (1) keyword based, (2) structural, and (3) semantic. Text search technology has matured at the keyword level, has made significant progress at the structural level (phrase and contiguity search, and XML-based search of structured documents), and is starting to push towards semantic search.

Applying text search technology to math search of equations and other math constructs faces serious problems, even at the keyword and structural levels. The first problem is that mathematical contents often involve symbols, as in |P_n(x)| and |d^2y/dx^2-x=0|, that are misinterpreted or unrecognized by text search systems. The second problem is that mathematical expressions have rich structures whose semantics are undetected by text search system. For example, |sin(x + log x)| is no different to a text search system than |sin x + log x|, and |x (y + z)| is misinterpreted as |x y + z|, if interpreted at all. The third problem is mathematical equivalence (``synonyms''). A sum or a product of several terms can be expressed in many equivalent ways; numbers can be represented in multiple forms (e.g., 1/2 vs. 0.5 vs. 2^-1); polynomials can be expressed in many factored and unfactored forms; and so on. Standard thesaurus-based approaches in text search are not adequate for searching for mathematically equivalent forms. The fourth problem is the issue of levels of abstraction in mathematical contents and queries. Should not a document containing f(x) match a query "f(t)", or a document containing f²+g² match a query "f(x)²+g(x)²"? The fifth problem is that of notational ambiguity in mathematics. Is dy the product d× y or the differential dy? Is x(t+s) the product x× (t+s) or the function x applied at t+s? Markup languages can eliminate ambiguity in the database contents, but not in users' queries.

In this talk, we will discuss the efforts and approaches for addressing those problems and, generally, for developing math search techniques and systems. At the keyword and structural levels of math search, two methodological strategies will be covered. The first is the evolutionary strategy of building a math search engine on top of a text search engine, and translating math contents (including the symbolic and the structural) in the database and in queries into textual counterparts for text search. The second strategy is to develop new schemes to index equations and structures directly, using parse trees and possibly other models, to increase the precision of searching. Common to both strategies is the necessity of an effective and easy-to-use interface for entering and editing math queries. The math search system being developed at NIST for the DLMF project will serve as an illustrative focus.

The talk will also cover pertinent aspects of the activities, projects, and products that are related to, or have impact upon, math search. These include math markup languages (OpenMath, MathMl, OMDOC), math ontologies and metadata, math languages/editors (EzMath, MathType, MathWriter, etc.), Math on the Web (MathWeb, MONET, the Esprit Project), and industry support.

Doron Zeilberger (Department of Mathematics, Rutgers University)  zeilberg@math.rutgers.edu

The General Future of Special Functions

I will survey the basic concepts building up to the holonomic systems approach to special functions identities, and provide an outlook on future prospectives and developments.

Material from Talks

Special Functions in the Digital Age

Foundations of Computational Mathematics 2002 (FoCM'02), August 5-14, 2002