Richard Askey: The Architect of Modern Special Functions (1933–2019)
Richard "Dick" Askey was a transformative figure in 20th-century mathematics. At a time when the study of "special functions"—the specific mathematical functions that solve physical and geometric problems—was considered by many to be a closed chapter of the 19th century, Askey single-handedly revitalized the field. His work bridged the gap between classical analysis, number theory, and physics, leaving behind a legacy that continues to influence quantum mechanics and coding theory today.
1. Biography: From St. Louis to the "Wisconsin School"
Richard Allen Askey was born on June 4, 1933, in St. Louis, Missouri. His academic journey began at Washington University in St. Louis, where he earned his B.A. in 1954. He then moved to Harvard University for his Master’s (1955) before completing his Ph.D. at Princeton University in 1961.
At Princeton, he studied under Salomon Bochner, a giant in harmonic analysis and probability. Askey’s dissertation focused on mean convergence of trigonometric series, setting the stage for his lifelong fascination with the behavior of functions.
After brief stints at Washington University and the University of Chicago, Askey joined the faculty at the University of Wisconsin–Madison in 1963. He would remain there for the rest of his career, eventually becoming the Gabor Szegő Professor of Mathematics. He was a cornerstone of the "Wisconsin School" of analysis, turning Madison into a global pilgrimage site for researchers interested in orthogonal polynomials.
2. Major Contributions: The Renaissance of Special Functions
Askey’s primary contribution was the modernization and systematization of special functions, particularly orthogonal polynomials.
- The Askey Scheme: Perhaps his most famous contribution is the "Askey Scheme of Hypergeometric Orthogonal Polynomials." This is a hierarchical classification system that organizes various families of polynomials (like Jacobi, Laguerre, and Hermite polynomials) based on their relationship to one another through limit transitions. It provided a roadmap for a field that had previously seemed like a disjointed collection of formulas.
- Askey-Wilson Polynomials: In collaboration with James Wilson, he discovered a highly general family of orthogonal polynomials. These "Askey-Wilson polynomials" are considered the "top" of the hierarchy in the Askey Scheme. They have profound applications in mathematical physics, particularly in the study of quantum groups.
- q-Series and Ramanujan: Askey was instrumental in bringing the work of the Indian genius Srinivasa Ramanujan back to the forefront of modern mathematics. He specialized in $q$-series (basic hypergeometric series), showing how Ramanujan’s seemingly "mystical" identities were actually part of a rigorous, structured framework of orthogonal polynomials.
- Positivity and Harmonic Analysis: He solved several long-standing conjectures regarding the "positivity" of certain sums of polynomials, which has practical implications in approximation theory and signal processing.
3. Notable Publications
Askey was a prolific writer known for his clarity and historical perspective. His most influential works include:
- "Orthogonal Polynomials and Special Functions" (1975): A SIAM monograph that served as a "manifesto" for the field, outlining the problems that would occupy researchers for the next three decades.
- "Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials" (1985): Co-authored with James Wilson, this paper introduced the Askey-Wilson polynomials and is one of the most cited works in the field.
- "Special Functions" (1999): Co-authored with George Andrews and Ranjan Roy, this 700-page tome published by Cambridge University Press is considered the definitive modern textbook on the subject.
- "Gabor Szegő: Collected Papers" (1982): Askey edited the works of his predecessor, adding extensive commentaries that contextualized classical results for modern readers.
4. Awards and Recognition
Askey’s work earned him the highest honors in the mathematical community:
- National Academy of Sciences (NAS): Elected as a member in 1999.
- American Academy of Arts and Sciences: Elected as a Fellow in 1993.
- Society for Industrial and Applied Mathematics (SIAM): He was a Fellow and a frequent invited speaker.
- Honorary Fellowships: He was a Fellow of the American Mathematical Society (AMS).
- The Gabor Szegő Prize: While he did not win it (as it is for early-career researchers), the prize was established in honor of the mathematician whose work Askey championed, and Askey was deeply involved in the community that oversees it.
5. Impact and Legacy
Askey’s legacy is twofold: mathematical and educational.
Mathematical Legacy
Before Askey, special functions were often viewed as "tools" for physicists rather than a field of pure mathematical inquiry. Askey proved they were central to the structure of mathematics. Today, his work is essential in Quantum Groups, Combinatorics, and Number Theory. The "Askey-Wilson algebra" is now a standard object of study in theoretical physics.
Educational Legacy
In his later years, Askey became a passionate advocate for improving K-12 mathematics education. He was a prominent voice in the "Math Wars" of the 1990s, arguing against "fuzzy math" and advocating for a curriculum that balanced conceptual understanding with rigorous basic skills. He frequently reviewed textbooks and testified before educational boards, insisting that children deserve to be taught mathematics that is both honest and deep.
6. Collaborations
Askey was a deeply social mathematician who thrived on collaboration.
- George Andrews: The world's leading expert on Ramanujan. Together, they explored the deep connections between combinatorics and special functions.
- James Wilson: His collaborator on the Askey-Wilson polynomials.
- Mourad Ismail: A long-time collaborator who helped expand the theory of $q$-orthogonal polynomials.
- Students: Askey mentored 15 Ph.D. students at UW-Madison, many of whom became leaders in the field, ensuring his "Wisconsin School" approach spread globally.
7. Lesser-Known Facts
- The "Human Encyclopedia": Askey was famous for his encyclopedic knowledge of mathematical history. He could often tell a researcher not only who had proved a theorem but also what they were thinking when they did it, often citing obscure papers from the 1800s.
- A Massive Library: He possessed one of the largest private collections of mathematics books in the world. After his death, his library—numbering in the thousands—was a testament to his bibliophilic devotion to the craft.
- The Ramanujan Connection: Askey played a crucial role in the publication of Ramanujan’s "Lost Notebook." He was one of the first to recognize that the "mock theta functions" Ramanujan wrote about on his deathbed were the keys to a new era of mathematics.
- Public Intellectual: He wasn't afraid of controversy. He once famously criticized a proposed California math framework so effectively that it led to significant revisions, earning him both fans and detractors in the world of pedagogy.
Richard Askey passed away on October 9, 2019. He is remembered not just for the polynomials that bear his name, but for his belief that mathematics is a continuous, living history where the old and the new are eternally linked.