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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 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 number theory, and the mathematics of Srinivasa Ramanujan. Mourad Ismail (see § -series,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 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.

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 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 §§ -Function.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.

In order to motivate and describe some of Askey’s deep mathematical contributions, it will be helpful to delve into the subject of and -analysis 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 -series because it was “highly contagious.” We are glad to see that the -disease” is here to stay. -disease

As Askey was one of the first to recognize, the subject of started with Fermat’s evaluation of -series where he replaced it by the infinite Riemann sum using the mesh points , that is, ,

which is now referred to as a When -integral. 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 , factorials -shifted

and the

respectively. We assume

The

To see the connection with partitions the reader should observe that the powers of

A basic hypergeometric function is a power series whose coefficients are quotients of products of

There are two divided difference operators associated with

has been used since the 19th century. The second is the Askey–Wilson operator

We note that if we think of the numerator as

There are many

where

Although many

### 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 * (1949, MR0030647). Here was the world of

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

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

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

In 1995, there was a two-week meeting on special functions,

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,

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

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

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

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

where *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

Thus, whenever the Askey–Wilson or

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

In their classification of

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

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

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 *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

A distance regular graph

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

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

We require

for

The Selberg integral

is the

Another important contribution is the Askey–Wilson integral

for

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

### 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

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

Askey’s papers (1973, MR0315351) and (1974, MR0372518) helped lead to the conjecture in Askey & Gasper AG76 that if