SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

OP-SF NET 3.3 (May 16, 1996), Topic #7

Waleed Al-Salam 1926-1996

Friends and colleagues are saddened by news of the passing, a few months short of his 70th birthday, of Waleed Al-Salam, Professor Emeritus of Mathematics at the University of Alberta. Born on July 15, 1926 in Baghdad, Iraq, he died on April 14, 1996 in Edmonton, Alberta, Canada.

Waleed studied at the University of California, Berkeley, earning a Bachelor's degree in Engineering Physics in 1950 and a M.A. in Mathematics in 1951. He returned to Baghdad as an Instructor at the College of Science for a few years, before enrolling for the Ph.D. at Duke University. He completed the degree in 1958 with a thesis "On the Bessel Polynomials" written under the supervision of Leonard Carlitz. By this time he was already a regular contributor to the periodical literature with some 20 published articles on a variety of topics in orthogonal polynomials and special functions.

After completing his Ph.D., Waleed returned to the College of Science in Baghdad as Associate Professor. Coming back to North America in 1962, he eventually (1966) took a position at the University of Alberta, where he was Professor of Mathematics from 1967 until his retirement in 1992. Waleed continued to contribute to several areas related to orthogonal polynomials; his CV lists over 80 articles. Areas covered by his work include characterization theorems (see his survey article in pp. 1-24 of P. Nevai, ed., Orthogonal Polynomials: Theory and Practice, Kluwer, 1990), Turan expressions, generating functions, summation formulas, q-analogs, and fractional operators. He and his collaborators did much work on various special and generalized systems of orthogonal polynomials. The Al-Salam-Carlitz polynomials (1965) are still frequently cited; see, e.g., the papers of R. Askey and S. K. Suslov in Lett. Math. Phys. 29 (1993), 123-132 and J. Phys. A 20 (1993), L693-L698. The Al-Salam-Chihara polynomials (1976) play an important role in the Askey-Wilson scheme of basic hypergeometric orthogonal polynomials.

Waleed supervised the Ph.D work of Bill Allaway (1972) and Mourad Ismail (1974), and conducted joint work with a variety of people at Alberta and elsewhere. In later years, these included Ted Chihara, A. Verma, Mourad Ismail and Waleed's wife, Nadhla Al-Salam, also on the faculty at Alberta, and a member of this Activity Group.

On his retirement in 1992, Waleed decided to put his expertise and energy at the disposal of the orthogonal polynomials community by starting and maintaining an ftp site for papers in the area. This effort prospered and he continued to oversee it until last year when his failing health made it necessary to pass the task to Hans Haubold at the UN Office in Vienna. Waleed had been diagnosed with leukemia in 1993 and this was to reduce his active participation in conferences in subsequent years. Nevertheless, with Nadhla's constant support, he still managed to travel and old and new friends were able to benefit from his knowledge and enjoy his optimistic and humorous personality.

Martin Muldoon

Remarks by Richard Askey

One of the pleasures in visiting Edmonton was being able to talk with Waleed Al-Salam. The conversations could be on mathematics, but just as frequently would be on other topics. What he did not tell me about one visit is something which shows his character.

It was a cold winter, and he met me at the downtown airport after a flight from Calgary. He never mentioned the sequel. I was wearing a fur hat bought in Moscow, and carrying a bag from the International Congress of Mathematicians in Moscow in 1966. There was a Russian hockey team in Calgary. When this Russian looking man with a Russian hat and bag flew to the downtown airport in Edmonton without the RCM having been notified, they called to Edmonton to ask that this be checked out. I did not notice them watching, nor do I think Waleed did, but a few days later there were questions asked about him and me at the Mathematics Department. It was only many years later that someone else told me about this. Waleed did not, since he did not want me to feel bad about his being investigated. That was typical. He was a gentleman.

Waleed was also a scholar. On another trip to Edmonton, I talked about some orthogonal polynomials which had first been found by L.J. Rogers, although Rogers was not aware they were orthogonal. These polynomials had been rediscovered twice around 1940, so were known to be orthogonal. However, they were not well known. In discussion after the talk, Waleed said that these polynomials reminded him of some polynomials found by Bill Allaway in his thesis. Waleed had sent me this thesis, but I had not read it carefully, so asked him to bring in a copy the next day. Sure enough, Allaway had rediscovered these polynomials once again, but had been more careful in his work than the two who had found these polynomials about 1940. There was a special case where division by zero occured unless you were careful, and Allaway had been. These polynomials which had been missed by others are now called sieved polynomials and they have played an important role in a few developments of the general theory of orthogonal polynomials. Both of Waleed's Ph.D. students, Bill Allaway and Mourad Ismail, got a good start in mathematics from the problems which they got from him.

I handled the paper Waleed and Ted Chihara wrote on what are now called the Al-Salam-Chihara polynomials. This was for SIAM Journal on Mathematical Analysis, so there should be some indication that the work would have applications to problems in science or engineering, or some other applied area. They had none, but the polynomials seemed so natural that there was no question in my mind that they would eventually have many applications. We now know where they live. They are a very important q-extension of Laguerre and Meixner polynomials. The problem Al-Salam and Chihara solved was a very natural extension of the problem solved by Meixner in the middle 1930s when he found Meixner polynomials and what I call Meixner-Pollaczek polynomials. These last arise in the representations of SU(1,1), and so in theoretical physics. Now that quantum groups seem to have appeared in physics, it is very likely that these polynomials of Al-Salam and Chihara will arise in problems in physics, just as the polynomials found earlier by Al-Salam and Carlitz have arisen in the study of q-harmonic oscillators. These last polynomials are natural q-extensions of Charlier polynomials. They also extend the continuous q-Hermite polynomials of L.J. Rogers. The polynomials of Rogers have a very interesting combinatorial structure associated with them, and extensions of this to the Al-Salam-Carlitz polynomials is currently being developed.

Waleed Al-Salam had a remarkable eye for formulas, and the ability and energy to find a number of important ones. The polynomials mentioned above are very appropriately named, and these and some other results found by him will keep his name in the minds of many people who never had the joy of knowing him. Those of us who did will remember him as well as his work, and miss him the more for this.

Richard Askey
Univ. of Wisconsin-Madison

Back to Home Page of
SIAM AG on Orthogonal Polynomials and Special Functions
Page maintained by Martin Muldoon