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The Legacy of Dick Askey (1933–2019)
Communicated by Notices Associate Editor William McCallum
1. History
Richard Allen (Dick) Askey, who devoted his life’s work to mathematics and mathematics education, died on October 9, 2019 at the age of 86.
Dick was born on June 4, 1933, in St. Louis, Missouri. In 1955, he earned a BA from Washington University in St. Louis, and in 1956 an MA from Harvard. He then pursued a doctorate at Princeton, and finished his course work in 1958. During 1958–1961, while completing his thesis, Dick was an instructor back at Washington University. In 1961, he earned a PhD from Princeton; his advisor was Salomon Bochner. After a two-year instructorship at the University of Chicago, Dick was appointed assistant professor in the department of mathematics at the University of Wisconsin, where he served for the remainder of his career. He became associate professor (1965–1968), professor (1968–1986), Gábor Szegő Professor (1986–1995), John Bascom Professor (1995–2003), and professor emeritus (2003–2019).
Dick Askey, Oberwolfach, March 1983.
Dick advised 14 PhD students and five postdoctoral fellows, all of whom thrived under his guidance. Later in this article, we will hear from his former PhD student Dennis Stanton (see §2.5), and his informal student/mentee Tom Koornwinder (see §2.8), who tell of their deep bond with Dick and his profound influence over them. Epic work with his former PhD student James Wilson is also featured. Dick’s later PhD students Shaun Cooper and Warren Johnson, and his later postdoc Frank Garvan, shared Dick’s interest in -series, number theory, and the mathematics of Srinivasa Ramanujan. Mourad Ismail (see §2.6) was one of Dick’s postdocs, and they developed a profound collaboration that continued throughout their careers. Dick also had a strong collaboration with George Gasper (see §2.7), who spent a year (1967–1968) at the University of Wisconsin as a visiting lecturer. Early in his career, Dick’s interest in the mathematics of Ramanujan brought him into contact with George Andrews (see §2.1), and they became lifelong collaborators. A shared interest in Ramanujan also brought Dick into contact with Bruce Berndt (see §2.3) who greatly benefited from Dick’s guidance for over four decades. Dick had collaborations with a large number of other mathematicians. For lack of space, we mention only a few: Mizan Rahman, Ranjan Roy, Paul Nevai, Deborah Haimo, Samuel Karlin, Natig Atakishiyev, Sergeĭ Suslov and Stephen Wainger.
Dick Askey, University of Minnesota, March 1988.
Dick was a preeminent mathematician of his generation, as the following awards and distinctions suggest. Dick was a Guggenheim Fellow (1969–1970); invited speaker at the International Congress of Mathematicians (1983); Vice President of the American Mathematical Society (AMS) (1986–1987); Honorary Fellow of the Indian Academy of Sciences (1988); Fellow of the American Academy of Arts and Sciences (1993); Member of the National Academy of Sciences (1999); Fellow of the Society for Industrial and Applied Mathematics (2009); and Fellow of the AMS (2012). Dick received an honorary doctorate from SASTRA University in Kumbakonam, India (2012), and a Distinguished Mathematics Educator Award from the Wisconsin Mathematics Council (2013). Dick won a Lifetime Achievement Award at the International Symposium on Orthogonal Polynomials, Special Functions and Applications in Hagenberg, Austria, on July 24, 2019.
Dick’s primary research interest was Special Functions; many of these are extensions of the functions on your scientific calculator. When asked why do research on special functions, Dick emphasized that one studies special functions not for their own sake, but because they are useful. Roughly speaking, special functions are the functions that have acquired a name after repeated use.
It took some courage for Dick to start his research career on the topic of special functions. During the period 1950–1970, it was widely believed that the existence of large, fast computing machines would minimize the value of special functions. This belief was wrong. Taking a broad view of the relationships between special functions and the rest of mathematics and physics, Dick and a small group of like-minded researchers resurrected the field and attracted many young, talented, and ambitious mathematicians to the area.
Dick was an author or coauthor of over 180 research articles. We mention two that had a profound influence. An inequality in his 1976 paper coauthored with George Gasper AG76 was used by Louis de Branges to prove the Bieberbach conjecture in 1985. In a Memoir published by the AMS in 1985 AW85, Dick and his former doctoral student, James Wilson, introduced the Askey–Wilson polynomials, which have become indispensable in combinatorics, probability, representation theory, and mathematical physics. The importance of these polynomials is suggested by the fact that the previously known families of hypergeometric and basic hypergeometric orthogonal polynomials, 43 families in total, are all special or limiting cases of the Askey–Wilson polynomials. The Askey–Wilson polynomials are viewed by many mathematicians as a sublime gift to their community.
Dick wrote two books, and he edited four more. His book, Orthogonal Polynomials and Special Functions Ask75 focused on classical orthogonal polynomials, related questions about positivity, and inequalities. His book, Special Functions AAR99, coauthored with George Andrews and Ranjan Roy, has become the standard text on the subject.
The elegance of Dick’s mathematical writing brings to mind the following quotation of Sun Tzu in The Art of War: The supreme art of war is to subdue the enemy without fighting. Many of Dick’s proofs have this quality.
First DLMF Editorial Board meeting, NIST, January 2000.
Dick lent his expertise to several projects that produced reference materials on special functions. In one project the National Institute of Standards and Technology (NIST) created the Digital Library of Mathematical Functions (DLMF); see NIST:DLMF. The DLMF is the 21st century successor to a classic text by Abramowitz and Stegun called the Handbook of Mathematical Functions (1964, MR167642). Dick served as an associate editor for the DLMF project. In this capacity, Dick gave advice on all aspects of the project, from its conception around 1995 to the initial release in 2010. In addition to his advising work, Dick coauthored the chapters on Algebraic and Analytic Methods, the Gamma Function, and Generalized Hypergeometric Functions & Meijer -Function. In another effort, Dick was involved in updating the Bateman Manuscript Project. The result is the Askey–Bateman Project in the Encyclopedia of Special Functions, edited by Mourad Ismail and Walter Van Assche, and published by Cambridge University Press. Volumes 1 and 2 have recently appeared, and they cover univariate/multivariate orthogonal polynomials along with some multivariate special functions that Dick was interested in (see §§2.2, 2.4, 2.5, 2.6, 2.8).
Dick was passionate about the history of mathematics, and he emphasized this topic in his lecturing and writing. Dick helped to edit A Century of Mathematics in America Ask89. Dick never tired of bringing to the world’s attention the genius of the mathematician Srinivasa Ramanujan (1887–1920). As part of this effort, in 1983 Dick commissioned the sculptor Paul Granlund to create a bronze bust of Ramanujan. Four copies were originally made, one of which is now in London at the headquarters of the Royal Society.
Early in his career, Dick made a commitment to improving K–12 mathematics education (see §§3.1, 3.2, 3.3). He wrote several dozen articles on this subject; for instance “Good intentions are not enough” Ask01. Dick was an advocate for the Singapore primary mathematics textbooks, and helped to create some of their Teacher Guides. Dick served on the AMS Education Committee (1998–2001) and the US National Committee for Mathematics (1999–2004). At the state level Dick consistently engaged in reviews and discussions concerning Wisconsin state standards, assessment documents, and professional resources. Dick’s mathematical credentials and common sense made him an effective critic of various fads in school mathematics education. Concerning his position, we will give him the last word:
Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps towards them at all levels.
Paul Terwilliger is a professor of mathematics at the University of Wisconsin–Madison. His email address is terwilli@math.wisc.edu.
2. Askey’s Contribution to Mathematics Research
Dick’s lifelong devotion to the study of special functions is summed up by this quote from Persi Diaconis:
Dear Friend Dick, You are one of my heroes. Not just because of your wonderful work but because of your bravery under fire. As we both know, there was a long time when our math world just didn’t know what to think about orthogonal polynomials: was it applied math, a corner of representation theory, or numerical analysis? Just what was it?? Anyway, it got “no respect.” You kept soldiering on and beat the ... at their own game.
Persi Diaconis and Dick Askey, 80th Birthday Conference, Madison, Wisconsin, December 2013.
In order to motivate and describe some of Askey’s deep mathematical contributions, it will be helpful to delve into the subject of -analysis and -series. As we will see, some -series serve as generating functions for statistics on partitions and also extend classical sums and integrals. The importance of looking at -series in a new and modern way started with Andrews and Askey’s work on orthogonal polynomials in the 1970s and suddenly many mathematicians quickly joined in. The spread of interest in -series was so fast that some called it “the -disease” because it was “highly contagious.” We are glad to see that the -disease is here to stay.
As Askey was one of the first to recognize, the subject of -series started with Fermat’s evaluation of , where he replaced it by the infinite Riemann sum using the mesh points , that is,
which is now referred to as a -integral. When the sum is a geometric progression and equals , which tends to as . This suggests considering the , , as analogues of the natural numbers and we are naturally led to the -shifted factorials
and the -binomial coefficients
respectively. We assume . One can then write , which then defines for .
The -binomial coefficient has many combinatorial interpretations. If is a prime power, it is the number of -dimensional subspaces of an -dimensional space over a field with elements. It is a polynomial in which is unimodal. It is also a generating function with power series variable of the number of partitions of integers which are of length and have at most parts. Many partition-theoretic identities have an analytic equivalence of the type “an infinite series involving -shifted factorials equals an infinite product.” For example the analytic forms of the famous Rogers–Ramanujan identities are:
To see the connection with partitions the reader should observe that the powers of on the right-hand side of the first equation are sums of terms of the form or , that is, partitions whose parts are (mod 5). Although the partition-theoretic interpretation of the left-hand side is not obvious, it can be thought of as the number of partitions into parts where any two parts differ by at least 2.
A basic hypergeometric function is a power series whose coefficients are quotients of products of -shifted factorials. The subject of -series studies evaluations of certain -series as quotients of infinite products, transformations connecting different -series, and orthogonal polynomials which arise as -series.
There are two divided difference operators associated with -analysis. The first,
has been used since the 19th century. The second is the Askey–Wilson operator . We write as and denote by . Then
We note that if we think of the numerator as , then the denominator will be exactly .
There are many -deformed physical models in which the Hamiltonian is a second-order linear operator in . One case is the -quantum invariant Heisenberg XXZ model of spin of a size with the open (Dirichlet) boundary condition. The Bethe ansatz equations of this model are
where . The solution of this system is identified with the zeros of a polynomial solution to a second-order equation in using a major modification of a technique which Stieltjes initiated to solve a one-dimensional electrostatic equilibrium problem. For details see §16.5 and §3.5 in Ismail (2009, MR2542683).
Although many -identities become hypergeometric identities as , there are many -results which exist only when . For example, the analytic form of the Rogers–Ramanujan identities 2 are genuine -series results and there is no limit.
2.1. Askey’s monographs
Dick edited many proceedings of conferences and collaborated with Madge Goldman and others on mathematics books for elementary school. He wrote two books for research mathematicians: Orthogonal Polynomials and Special Functions Ask75 (stemming from his 1974 Conference Board of the Mathematical Sciences (CBMS) lecture series at Virginia Tech), and Special Functions AAR99 (joint with Ranjan Roy and George Andrews). Dick’s editing and comments on Gábor Szegő’s collected papers in three separate volumes Sze82 has been immensely valuable.
Orthogonal Polynomials and Special Functions is an elegant introduction to Dick’s early achievements, but more importantly, also to the topics that he viewed as most significant. He and I had been in correspondence since 1970, and he invited me as one of the participants in the CBMS conference. It was about then that our two quite different fields of interest converged. I recall saying to Dick that before I met him, I didn’t know an orthogonal polynomial from a perpendicular one. He responded, “I’d hate to tell you what I thought a partition was!” The confluence of our interests led Dick to invite me to Wisconsin for the 1975–1976 academic year. Thus began the long path leading to the book, Special Functions. We decided to run a seminar on our joint interests. As a topic, we chose Wolfgang Hahn’s paper, Ueber Orthogonalpolynome, die -Differenzengleichungen genuegen (1949, MR0030647). Here was the world of and orthogonal polynomials tied neatly together. We decided we should definitely write a book. I set to work to produce three chapters, and Dick was collecting notes from our seminar to supplement his contribution. Each of us got enmeshed in other projects after the glorious year of 1975–1976. Cambridge University Press continued to nudge us over the decades, and we intended, over and over again, to put everything together. As time wore on, it seemed this book would never happen. The late Ranjan Roy was entirely responsible for rescuing it from oblivion. Ranjan attended subsequent lectures by Dick and realized that the long-awaited book would never come about unless someone took over the onerous task of taking the rough notes and ideas from both of us and putting them into a coherent and readable text. It was clear from Ranjan’s other singly authored books that he was an excellent mathematical expositor, and hence he was the perfect choice for our book. Both Dick and I were grateful beyond words that Ranjan was able to turn our 1976 dream into a 1999 reality.
Andrews, Askey, and Roy, Baltimore, 2003.
George E. Andrews is the Evan Pugh Professor of Mathematics at Pennsylvania State University. His email address is gea1@psu.edu.
2.2. Askey and Ramanujan
Richard Askey was widely acknowledged as a leader in the field of special functions. He was also a major figure in the world of Ramanujan, for he was instrumental, along with George Andrews and Bruce Berndt, in making the mathematical world aware of the wide-ranging and deep contributions of Ramanujan. Indeed, he, George Andrews, and Bruce Berndt have been jocularly referred to as “the gang of three” in the Ramanujan world. Here I shall share some personal recollections, but in doing so, I shall focus on Askey’s role in educating us about the remarkable contributions of Ramanujan pertaining to special functions, and in his efforts to foster the legacy of Ramanujan.
I first met Askey at the Joint Summer Meeting of the AMS and MAA at the University of Michigan, Ann Arbor, in August 1980. At that time, I was a Hildebrandt Research Assistant Professor there just after having completed my PhD. He was giving the J. Sutherland Frame Lecture on “Ramanujan and some extensions of the gamma and beta functions” which I attended. I was charmed by his conversational, yet engaging, lecturing style. A vast panorama of the area of special functions unfolded in his lecture, revealing his encyclopedic knowledge of the subject. He discussed some of Ramanujan’s startling discoveries and exhorted everyone in the audience to study the work of the Indian genius. In particular, he emphasized Ramanujan’s important -analog of the beta integral. He also discussed Selberg’s multidimensional extension of the beta integral, and spoke about his conjectured -analog of the Selberg integral, which provided an extension of the integrals of both Ramanujan and Selberg. He concluded his lecture by pointing out that in physics there are incredible formulas in several variables that are being analyzed and that a genius like Ramanujan would be of invaluable help. As Askey put it:
The great age of formulae may be over, but the age of great formulae is not!
Askey’s paper under the same title as the lecture had just appeared in the May 1980 issue of the American Mathematical Monthly. I was working in analytic number theory at that time, but before the end of that decade, owing to the lectures of Andrews, Askey, and Berndt that I heard at the Ramanujan Centennial in India in 1987, I entered the world of , or as Askey would say, I was smitten with the -disease!
Around the time of that Summer Meeting, Askey sent letters to academicians and persons with an interest in fostering scientific legacies to contribute towards the creation of busts of Ramanujan. I received one such letter. Responding to a plea from Janaki, Ramanujan’s 80-year-old widow who was living in poverty in Madras, India, Askey had contacted the famous American sculptor Paul Granlund and commissioned him to produce busts based on the famous passport photograph of Ramanujan. The response to Askey’s letter of request was overwhelming, and so it was possible for Granlund to produce ten bronze busts, and these were ready by 1983, well in time for the Ramanujan Centennial in 1987.
The Ramanujan Centennial was an occasion when mathematicians around the world gathered in India to pay homage to the Indian genius, and take stock of the influence his work has had and the impact it might have in the future. Askey was one of the stars of the centennial celebrations. There were several conferences in India during December 1987–January 1988, and Askey was a speaker in almost all of them. I organized a one-day session during a conference at Anna University, Madras, in December 1987. He graciously accepted our invitation to inaugurate that conference and to speak in my session. Mrs. Janaki Ramanujan was present at the inauguration, and she thanked Askey profusely for his effort in getting the busts of Ramanujan made. After the inauguration, Askey delivered a magnificent lecture on “Beta integrals before and after Ramanujan” in my session. We were also honored to have him give a public lecture entitled “Thoughts on Ramanujan” at our family home in Madras under the auspices of the Alladi Foundation that my father, the late Prof. Alladi Ramakrishnan, had created in memory of my grandfather Sir Alladi Krishnaswami Iyer.
With my research focused on the theory of partitions and -series since 1990, we have had a series of conferences at the University of Florida emphasizing this area. Professor Askey has visited Gainesville several times both as a lead speaker at these conferences, and for History Lectures and talks on mathematics education during the regular academic year. I have enjoyed every one of his lectures in Gainesville and at meetings elsewhere. I want to share with you one interesting episode.
In 1995, there was a two-week meeting on special functions, -series, and related topics at the Fields Institute in Toronto. The first week was an instructional workshop, and the second week was a research conference. I attended the second week. The great I. M. Gel’fand was scheduled to be the Opening Speaker for the research conference. I was looking forward to Gel’fand’s lecture since I had heard so much about the Gel’fand Seminar he had conducted in Moscow, and how he would dominate the seminar and cut people down to size. It turned out that Gel’fand could not come to Toronto due to ill health (he was 82 years old). So Askey got up and said that he was the one who had invited Gel’fand, and if the person you had invited is unable to come, then you should give a talk in his place. So in his imitable style, Askey gave a masterly lecture on special functions that I thoroughly enjoyed.
His insight and critical comments have been immensely useful to me in various ways. Starting from the Ramanujan Centennial, I wrote articles annually for Ramanujan’s birthday for The Hindu, India’s National Newspaper, comparing Ramanujan’s work with that of various mathematical luminaries in history. I benefited from Askey’s comments and (constructive) criticism in preparing these articles. A collection of these articles appeared in a book that I published with Springer in 2012 for Ramanujan’s 125th birthday.
By the time the Ramanujan 125 celebrations came around in 2012, I was firmly entrenched in the Ramanujan World, and so was involved with the celebrations in various ways. In particular, owing to my strong association with SASTRA University, I organized a conference at their campus in Kumbakonam, Ramanujan’s hometown. We felt that Askey, Andrews, and Berndt had to be recognized in a special way in Ramanujan’s hometown for all they had done to help us understand the plethora of identities that Ramanujan had discovered. So The Trinity (Askey, Andrews, and Berndt)—as I like to refer to them in comparison with the three main Gods, Brahma, Vishnu, and Shiva of the Hindu religion (!)—were awarded Honorary Doctorates by SASTRA University in a colorful ceremony at the start of which they entered the auditorium with traditional South Indian Carnatic music being played on the Nadaswaram, a powerful wind instrument. Askey enjoyed the ceremony but felt that the music was too loud; in fact, that is how the Nadaswaram is, since it is played in festivals attended by a thousand people or more!
There is much that can be said of Dick Askey. But I will conclude by emphasizing that, in spite of his eminence, he was a very friendly and helpful person. It is rare to find eminence combined with humanity, and Askey had this precious combination which has been beneficial to so many of us. In particular, we owe a lot to him for helping us understand some aspects of Ramanujan’s fundamental work on special functions, and for his efforts in fostering the legacy of the Indian genius.
Krishnaswami Alladi is a professor of mathematics at the University of Florida. His email address is alladik@ufl.edu.
2.3. Askey and Ramanujan’s notebooks
Although I was a graduate student at Wisconsin, and Richard Askey was a close friend and strong supporter of my work for over 50 years, one of my life’s biggest regrets is that I never took a course from him. My first experiences with number theory came in the spring semester of my third year and fall semester of my fourth year at Wisconsin when I enrolled in courses in modular forms, taught, respectively, by Rod Smart and Marvin Knopp. Perhaps surprisingly, modular forms led me to a doctoral dissertation in which Bessel functions played a leading role. My association with Dick and his advocacy of my research for the next 53 years began at this time.
While on my first sabbatical leave at the Institute for Advanced Study in February 1974, I discovered that theorems that I had proved on Eisenstein series at the Institute enabled me to prove some formulas from Ramanujan’s notebooks. Starting in May 1977, I began devoting all of my research efforts for the next forty years to proving the claims made by Ramanujan in his (earlier) notebooks and later, with George Andrews, in his lost notebook. It is unfortunate that there is not sufficient space here in which to express my appreciation and indebtedness to my many doctoral students who enormously aided me in this long endeavor. By far, the most important and strongest advocate of not only my work, but also that of my students, during these several decades was Dick Askey.
During my efforts to find proofs of Ramanujan’s claims in his notebooks Ram57, Askey provided many insights, references, and proofs. In my five volumes (abbreviated by Parts I–V), devoted to Ramanujan’s claims, I referred to Askey’s help a total of 31 times, more than any other mathematician. (For brevity of the present exposition, complete references to Askey’s several relevant papers can be found in Ber85.) We now provide a sampling of Askey’s contributions to our editing of Ramanujan’s notebooks Ram57.
Already in Part I, published in 1985, Askey read in detail most of the chapters, and, in particular, supplied many important references. In generalizing a result of Ramanujan Ber85, p. 302, Askey showed that a more general integral
using 1, , , is a -analogue of the beta integral 3.
As most friends of Askey are aware, he had a prodigious knowledge of the literature, especially that from the 19th century and earlier. The chapters in Part II are significantly richer because of Askey’s historical observations. In Chapter 11, which features hypergeometric series, Askey’s influence is most pronounced, as he supplied several proofs and observations. For example, in Entry 29(ii) of Chapter 11 in his second notebook Ram57, pp. 86–87, Ramanujan offered an identity for hypergeometric functions, namely,
where , or is a nonnegative integer. Askey and Wilson showed that this identity leads to an orthogonal set of polynomials on with respect to a weight function involving a product of gamma functions.
The Rogers–Ramanujan identities 2 appear in Part III Ber85, pp. 77–79, where a lengthy discussion of all known proofs up to 1991, including a new proof from Askey, can be found. On page 284 in his second notebook, Ramanujan writes,
The difference between and
but he does not tell us what the difference is. We might guess that it is 0, and it is in some cases. Askey provided the answer, which you can find in Ber85, Part IV, pp. 344–346.
Two of the most intriguing entries in the 100 pages of unorganized material in Ramanujan’s second notebook pertain to Gaussian quadrature. On pages 349 and 352 of Ram57, Vol. 1 Ber85, Part V, pp. 549–560, Ramanujan provides theorems on Gaussian quadrature with respect to a discrete measure in which orthogonal polynomials play a central role. We provide a portion of one example. Let
(Ramanujan did not provide any hypotheses for ) Ramanujan then gives four successive approximations to , the first of which is simply , and the second is
Askey pointed out that an application leads to Hahn polynomials, which were introduced by Chebyshev in 1875 and are constant multiples of
The proofs are due to Askey and do not apparently appear elsewhere. As Askey pointed out, it was surprising to learn of these theorems, because nowhere else in Ramanujan’s work is there any indication that he knew about Gaussian quadrature and orthogonal polynomials.
Although he would not acknowledge such an acclamation during his time, Askey was generally recognized as the world’s leading authority on -series, which flow abundantly throughout Ramanujan’s lost notebook Ram88. The advice and comments on -series that Askey kindly gave to George Andrews and this writer in preparing our five volumes on Ramanujan’s lost notebook AB05 cannot be overemphasized.
Ramanujan and Dick Askey would have immensely enjoyed having long conversations with each other.
Bruce C. Berndt is a professor of mathematics at the University of Illinois at Urbana-Champaign. His email address is berndt@math.illinois.edu.
2.4. Askey and algebra
Even though an algebra now bears his name, Askey himself was not very involved with algebraic studies. In fact, he often told us that the nomenclature to which we shall refer should be changed so as to not mention him. It is however a striking manifestation of his legacy that his work has led to constructs which are becoming more and more important in Algebra and Mathematical Physics.
Empirically, it is observed that the possibility of solving physical models rests on the underlying presence of symmetries that can be somewhat hidden. Their mathematical description has led to the identification of structures such as Lie algebras, superalgebras, quantum algebras, and their representation theory. The special functions that appear in the solutions must thus offer a lead as to what are the algebraic entities poised to account for the symmetries of the systems in question. This relates to the long tradition championed by Wigner, Gel’fand, Vilenkin among others of interpreting special functions algebraically.
What is then the algebra encoded in , the Askey–Wilson polynomials AW85, and their corresponding finite set, the -Racah polynomials? The answer (to which we have contributed) is rooted in the bispectral properties of the Askey–Wilson polynomials which are eigenfunctions of a -difference operator in addition to satisfying, as required for orthogonal polynomials, a three-term recurrence relation where the variable can be viewed as the eigenvalue of an operator acting on the discrete degree variable. Focusing on these two operators and setting and , the following relations are found:
where and , , and are related to the parameters , , , of the polynomials . Since the realization is not affected by the truncation, the algebra is also the one associated to the -Racah polynomials. Focusing on generators and relations, this can be taken to define the Askey–Wilson algebra abstractly. It is remarkable that this algebra encapsulates the properties of the Askey–Wilson polynomials which can indeed be obtained from the construction of representations. An interpretation of the polynomials as overlaps between the eigenbases of and follows. This relates to the theory of Leonard pairs and to - and -polynomial association schemes, both already mentioned (see §2.5).
Thus, whenever the Askey–Wilson or -Racah polynomials are present, the Askey–Wilson algebra is lurking. Now it is known that the -Racah polynomials are basically the -coefficients of the quantum algebra . Such coefficients arise in the recouplings of three irreducible representations. This suggests, as is the case, that the Askey–Wilson algebra occurs as the centralizer of the diagonal action in representations of the three-fold product of . Here the generators are realized as the intermediate Casimir elements and the parameters are related to the values of the initial and total Casimir operators. The Askey–Wilson algebra is ubiquitous: it is a coideal subalgebra of , a truncation of the -Onsager algebra, it is connected to double affine Hecke algebras (DAHA), it identifies with the Kauffmann bracket skein algebra of a four-punctured sphere, offers a framework to extend Schur–Weyl duality, and so on. We are much endebted to Askey for all that.
The bispectral properties of the Racah polynomials can similarly be packaged in an algebra to which the name of Racah has been attached. It can be obtained as the limit of the Askey–Wilson algebra after an affine transformation of the generators has been performed to revert to ordinary commutators. This Racah algebra is the centralizer of the diagonal action of in its three-fold tensor product and is the symmetry algebra of the generic superintegrable model in two dimensions.
In their classification of - and - polynomial association schemes, Bannai and Ito found a case that corresponds to the limit of the -Racah polynomials which are now referred to as the Bannai–Ito polynomials. We observed (with S. Tsujimoto) that these are eigenfunctions of a certain Dunkl shift operator. The corresponding eponymous algebra proved to be the centralizer of three copies of the superalgebra . This led to the characterization of a “ scheme” complementing the Askey tableau. Askey took an interest in these studies, and on numerous occasions, he expressed the view that it should be possible to extend this to other roots of unity.
There is much more that Askey has generated in Algebra with the introduction of the Askey–Wilson polynomials. For instance, a noteworthy conduit has been their multivariate generalizations. Two directions have been followed in this respect. One, in the framework of symmetric functions, led to the Macdonald–Koornwinder polynomials corresponding to the root lattice. DAHAs are in this case the associated algebraic structures. The other took the recoupling path with Tratnik, Gasper, and Rahman providing different generalizations of the Racah and Askey–Wilson orthogonal polynomials in many variables. The extensions of the Askey–Wilson, Racah, and Bannai–Ito algebras that these last classes of polynomials entail are currently being developed.
The influence of Askey on Algebra will endure. To make clear that there are many more dimensions to this impact, we may add that the other Askey polynomials, those defined on the circle which are biorthogonal (see Sze82, Vol. 1), have been connected to the Heisenberg group and it is expected that more algebraic advances will arise from the exploration of these functions. This is another illustration that Askey’s results will keep cross-fertilizing areas of representation theory, special functions, and mathematical physics in ways that have not yet been fully imagined.
Luc Vinet is Aisenstadt Professor at the Centre de Recherches Mathématiques, Université de Montréal. His email address is luc.vinet@umontreal.ca.
Alexei Zhedanov is a professor of mathematics at the Renmin University of China. His email address is zhedanov@ruc.edu.cn.
2.5. Askey and combinatorics
Askey considered enumerative questions on Laguerre polynomials with Ismail (1976, MR0406808) and also Ismail–Koornwinder (1978, MR0514623). Some integrals could be evaluated by counting certain permutations, and weighted versions with parameters also existed. Even–Gillis (1976, MR0392590) had considered similar questions. One of their methods was the MacMahon Master Theorem in enumeration. At the same time, Foata (1978, MR0498167) had used the exponential formula in enumeration to give a beautiful proof of Mehler’s formula for Hermite polynomials
This began a good deal of work in enumeration and specific orthogonal polynomial systems, orchestrated by Askey and Foata. They were instigators for important conferences in Columbus, Oberwolfach, and Tempe for researchers in both areas. At this time -analogues of classical polynomials were intensely studied, leading to connections to partition theory and quantum groups. The philosophy of using weighted objects to represent analytic statements carried over to general orthogonal polynomials, and was developed by Flajolet (1980, MR0592851) using continued fractions, and by Viennot (1985, MR0838979) using combinatorial techniques. This combinatorial philosophy was used by Zeilberger–Bressoud (1985, MR0791661) to prove the -Dyson conjecture.
Askey realized that a finite set of orthogonal polynomials with a finite discrete orthogonality always has a dual orthogonality. He reorganized the orthogonality relation as row or column orthogonalities of an orthogonal matrix. The symbols had such orthogonalities. Askey and Wilson (1979, MR0541097) showed these symbols, once rescaled, were orthogonal polynomials in one variable. They are part of the classical scheme of hypergeometric orthogonal polynomials, namely functions with four free parameters. Both orthogonalities could be reformulated for polynomials. Askey made a partially ordered set which organized the classical hypergeometric polynomials (e.g., Hermite, Laguerre, Charlier, Jacobi, Meixner, Krawtchouk, Racah), the tableau d’Askey. The Hasse diagram of this poset (partially ordered set), drawn and distributed by J. Labelle (1984, MR0838967), became a focus of study. The -Racah polynomials, defined by basic hypergeometric series, were the top element of the discrete part of this diagram, and would be key polynomials in algebraic graph theory.
A distance regular graph (see Bannai–Ito, 1984 MR0882540) has very regular properties. Among them is that the indicator matrix for vertices at distance in the graph is a polynomial of degree in the distance one matrix If this polynomial is denoted , the polynomials have a discrete orthogonality relation using the values of where are the eigenvalues of Delsarte (1973, MR0384310) studied a special case of these graphs called - and -polynomial association schemes, for which there are polynomials and real numbers satisfying These have two sets of orthogonalities, one for and one for , just as the Racah polynomials did. Askey knew that all of the known infinite families of such schemes had eigenmatrices given by special or limiting cases of the -Racah polynomials. Leonard (1982, MR0661597) proved the surprising result that the eigenmatrices are always special or limiting cases of the -Racah polynomials. Wang (1952, MR0047345) classified the two-point homogeneous spaces, whose spherical functions, the continuous versions of are classical orthogonal polynomials. This gives some hope to classify such association schemes. Terwilliger (2001, MR1826654) has developed a detailed study of the linear algebra behind the pair of matrices for and .
Askey (1975, MR0481145) promoted the study of linearization and connection coefficient problems for orthogonal polynomials. The linearization coefficients had graph- and group-theoretic interpretations, which could be restated as an enumeration problem. Rogers’ original proof of the Rogers–Ramanujan identities 2 used a connection relation for the -Hermite polynomials to the -Hermite polynomials. Rogers had these polynomials, but did not know their orthogonality relation.
Askey considered as crucial the integrals or sums for the total mass of the weight function for orthogonal polynomials. For the Jacobi polynomials, this integral is a beta function. The Askey–Wilson integral (1985, MR0783216) 5 is the evaluation for the continuous weight at the top of the tableau d’Askey. Askey was keenly interested in multivariable and root system versions (1980, MR0595822), and organized work in this area. His student Walter Morris (1982, MR2631899) wrote a thesis with many such conjectured integrals for root systems, concurrent with Macdonald (1981, MR0633515). These became important as measures for orthogonal polynomials in several variables; Macdonald (1989, MR1100299) and Koornwinder (1992, MR1199128).
Dennis Stanton is a professor of mathematics at the University of Minnesota. His email address is stanton@math.umn.edu.
2.6. Askey and beta integrals, by Mourad E. H. Ismail
One of Dick Askey’s contributions is the insight to see that many -series identities, old and new, are different -analogies of the beta integral
We require . One remarkable example is the Ramanujan sum
for ; see AAR99, §10.5 for details.
The Selberg integral
is the -dimensional version of the beta integral, which corresponds to the case . Dick recognized that the Selberg integral is the key to the development of a deep theory of multivariate special functions. He also formulated -analogues of this integral and of Aomoto’s generalization of the Selberg integral. Askey also promoted the work of I. G. Macdonald and others on root systems. The last forty years saw great progress in this area spearheaded by Dick’s tireless promotion and encouragement. This eventually led to the theory of Macdonald and Koornwinder polynomials. The interested reader may consult the beautiful survey of the Selberg integral, its applications, and significance by Peter Forrester and Ole Warnaar (2008, MR2434345).
Another important contribution is the Askey–Wilson integral
for . The orthogonality of the Askey–Wilson polynomials follows from 5 in a standard way. This again can be interpreted as a -beta integral. After a certain scaling and letting it becomes the Wilson integral
valid for .
The Wilson integral and the Wilson polynomials appeared first in Wilson’s dissertation written under Askey’s supervision. The Askey–Wilson integral is a fundamental result that led to a better understanding of -special functions, their transformations, and analytic properties.
2.7. Askey, positivity, inequalities, and applications
In the spring of 1967, shortly after accepting a visiting lecturer position at the Mathematics Department of the University of Wisconsin in Madison, I received a package from Professor Richard Askey containing several interesting reprints and preprints of his papers (partially joint work with Isidore Hirschman, Jr., Steve Wainger, and Ralph Boas) and a letter encouraging me to attend his graduate-level special functions course at the university during the 1967–1968 academic year. So, in addition to teaching the graduate complex variables course and attending analysis seminars, talks, and Wainger’s harmonic analysis course, I also attended Askey’s special functions course.
Dick’s course covered gamma and beta functions, generalized hypergeometric functions and series, Bessel functions, (mostly classical) orthogonal polynomials (and their three-term recurrence relations, differential/ difference equations, asymptotic expansions, etc.), summability, fractional integrals, infinite products, etc. You can now study most of the topics covered in the course by reading the corresponding material in the 1999 Andrews, Askey, and Roy Special Functions book AAR99. Dick was an excellent, knowledgeable lecturer who communicated enthusiastically with his audiences and was able to get them involved in the discussions. With his prodigious memory he could lecture on several mathematical topics and rapidly write complicated formulas on the blackboards without referring to his notes. At the beginning of classes he would frequently pass out stapled piles of blue mimeographed sheets containing the formulas, definitions, theorems, etc., that he was going to discuss. Also, in his classes and talks he would point out related open problems that he and others had tried to solve, and strongly encouraged his audience to try to solve them. It was Askey’s encouragements to solve interesting and important open problems that led to many significant papers.
Via the positivity of the generalized translation operator for Jacobi series in Gasper (1971, MR0284628), Askey (1972, MR0340672) proved that if and , where , , then in order for all of the partial sums of the Poisson kernel in powers of , for Jacobi series to be nonnegative for , it is necessary and sufficient that He applied Bateman’s fractional integral and some identities and inequalities for Jacobi polynomials to show that the inequality displayed above holds when . In (1972, MR0301897) and (1979, MR0539375) he applied this inequality and an inequality for sums of Jacobi polynomials in Gasper (1977, MR0432946) to prove that the Cotes numbers for Jacobi abscissas are positive if , or or if . He also stated that it should be possible to fill in the convex hull of these points and those in an earlier paper with Fitch (1968, MR0228166) and that it is possible that the Cotes numbers are positive on the rectangle which would be the best possible rectangle, both of which are still open problems.
In a paper AG76 that was submitted for publication in 1973 and published in 1976, Askey and Gasper used a sum of squares of ultraspherical polynomials and Bateman’s fractional integral to prove that the sum of Jacobi polynomials displayed above is nonnegative for , , , if and only if Unexpectedly, several years later the special cases of the above inequality turned out to be the inequalities that de Branges (1985, MR0772434) needed in February 1984, to complete his proof of the Bieberbach conjecture, and of the more general Robertson and Milin conjectures. For additional information, see the Askey and Gasper paper, Askey’s personal account, and the other papers and personal accounts in DDM86.
Askey’s papers (1973, MR0315351) and (1974, MR0372518) helped lead to the conjecture in Askey & Gasper AG76 that if