SIAM News Blog

A SIAM Special Function at NIST

By Willard Miller, Jr.

The 13th International Symposium on Orthogonal Polynomials, Special Functions and Applications was held at the National Institute of Standards and Technology, Gaithersburg, Maryland, June 1-5, 2015. OPSFA-13 was jointly sponsored by the SIAM Activity Group on Orthogonal Polynomials and Special Functions and the NIST Applied and Computational Mathematics Division, with support from the National Science Foundation and NIST. NIST, formerly called the National Bureau of Standards, was a particularly appropriate host for the symposium (the first held in the U.S.): NBS/NIST has been a leader in the development of handbooks and other reference materials that provide basic formulas and notation for special functions, as well as numerical tables and software.

There is no technical definition of special functions. Basically, they are useful functions, occurring so frequently in the sciences that it becomes imperative to collect them in handbooks, with descriptions of their properties and codifying notation. Most of these functions are the “special functions of mathematical physics” (arising as solutions of second-order linear differential and difference equations in chemistry, physics, and engineering, perhaps via separation-of-variables methods or as families of orthogonal polynomials); examples include hypergeometric and \(q\)-hypergeometric functions. More recently, Painlevé functions (satisfying second-order nonlinear differential equations), exceptional polynomials (the Askey scheme and beyond), and other functions have risen to handbook status. 

NBS/NIST has played an invaluable role in codifying and disseminating information about special functions. Work on the NBS tables started in 1938, culminating in the 1964 publication of the NBS handbook (edited by Milton Abramowitz and Irene A. Stegun). The earlier NBS tables were largely numerical, oriented toward practical computations. In recognition of the emerging predominance of mathematical software, the 1964 handbook focused more on formulas, with tables of numerical values filling fewer than half its pages.

Walter Gautschi and the late Frank Olver played major roles in developing the material for the handbook, of which more than a million copies are probably in print. It remains the most highly cited mathematics publication of NIST.

Karl Liechty received the Szegö Prize "for his original work in the asymptotic analysis of orthogonal polynomials arising in models from statistical mechanics, in particular the six-vertex model and a model of non-intersecting random paths.” Photo by Walter Van Assche.
In 2010 the Digital Library of Mathematical Functions succeeded the 1964 handbook; DLMF is available as a handbook and also online. Almost no tables of numbers appear in DLMF, which contains more than twice as many formulas as the old handbook. The online version is continually updated; it imparts information about the behavior of these functions through interactive graphics. Frank Olver provided leadership on subject matter for the project (he was the founding editor of SIAM Journal on Mathematical Analysis (1970), which before a change in editorial policy was among the most prestigious journals covering special functions.) The other DLMF editors are Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark. Part of this year’s symposium was devoted to assessments of the history and continuing updates of the project, and discussions of plans for the future.

The OPSFA International Symposia date back to 1984. At the first meeting, held in Bar-Le-Duc, France, the invited speakers were the luminaries J. Dieudonné, “Fractions contiuées et polynômes orthogonaux dans l’œuvre de E.N. Laguerre”; W. Hahn, “Über Orthogonalpolynome die linearen Differenzengleichungen genügen”; G. Andrews and R. Askey, “Classical orthogonal polynomials”; and W. Gautschi, “Some new applications of orthogonal polynomials.”

The community of researchers who use special functions is divided into two overlapping categories: those whose main interest is the special functions themselves and their properties, and those who are motivated by other branches of the sciences but encounter special functions and orthogonal polynomials in these pursuits. This meeting was no exception. Four of the ten plenary talks focused on the core body of the theory of special functions and orthogonal polynomials: 

  1. Charles Dunkl: vector-valued Jack and Macdonald polynomials
  2. Mourad Ismail: \(q\)-polynomials
  3. Nico Temme: asymptotic and computational aspects of special functions
  4. Teresa Pérez: multivariate orthogonal polynomials and moment functionals. Other research areas furnished the motivation for the remaining six
  5. Percy Deift: random matrices, Riemann-Hilbert problems, Airy functions
  6. Wadim Zudilin: number theory, Riemann zeta function
  7. Craig A. Tracy: random matrices
  8. Sarah Post: symmetries in mathematical physics
  9. Olga Holtz: polynomials with real roots, optimization theory
  10. Lauren Williams: models of particles on lattices. 

View the full program here.

Every two years SIAG/OPSF, joint with SIAM and the OPSFA symposium series leadership, awards the Gábor Szegö Prize to an early-career researcher for outstanding research contributions in the area of orthogonal polynomials and special functions. This year the prize was awarded to Karl Liechty of DePaul University (see photo), who delivered the Gábor Szegö Prize lecture: “Tacnode Kernels and Lax Systems for the Painlevé II Equation.” 

The SIAM Activity Group on Orthogonal Polynomials and Special Functions was founded in 1990, under the leadership of Charles Dunkl. The group’s newsletter has appeared since 1993, and the Gábor Szegö Prize was launched in 2010, thanks to then SIAG/OPSF chair Francisco Marcellán.

In this era of machine computation and computer simulation, special functions and orthogonal polynomials remain vibrant areas of research for reasons far beyond the intrinsic beauty of the theory. The construction of realistic mathematical models of real-world systems that can be solved analytically and explicitly with adjustable parameters through the use of special functions plays a vital role in the understanding of the structure of these systems. This role was clearly demonstrated at OPSFA-13.

Willard Miller, Jr., is a professor emeritus of the School of Mathematics at the University of Minnesota. 

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