Dick Askey: A Special (and VERY IMPORTANT!) Guru Indeed

Doron ZEILBERGER

Special Functions are functions that occur so often that they deserve a name, but even more important than functions are people, and Dick Askey is one of the most special people I have ever met.

In this brief note, let me mention a small sample of the very significant mathematics that he inspired.

• It was Dick who challenged Dominique Foata to find a combinatorial proof of the Mehler formula, and that lead to Foata’s beautiful combinatorial approach to Special Functions, pursued by him, his collaborators, and many others.

• It was Dick who had brilliant PhD students, including Dennis Stanton, that became the absolute unit of quality for all PhD students. Being x-Dennis with x > \frac{1}{2} means that the student is very good. Of course x ≤ 1, and the inequality is sharp, with only one case (guess who?) achieving x = 1.

• Dick is very passionate about mathematics education, something very rate amongst research mathematicians, and is a ‘gadfly’ (with co-gadfly George Andrews) about what they deemed misguided reforms in K-12 education, stating that Good intentions are not enough. Dick did not just criticize, but set an example by writing insightful articles for the Mathematics Teacher, the leading periodical for high-school teachers, about Fibonacci and Lucas numbers.

• The ‘Askey scheme’ (aka tableau d’Askey), hanging in my office (nicely designed by Jacques Labelle), was the conceptual skeleton of lots of great research in special functions, highlighting the Askey hypergeometric hierarchy, with the Racah and Askey-Wilson polynomials on the top.

• Dick caught the q-disease, and along with George Andrews and others (Mizan Rahman, George Gasper, Dennis Stanton, Frank Garvan to name a few) inspired lots of insightful q-analogs of classical theorems, that lead to insightful combinatorial interpretations (Xavier Viennot and his school, Bill Chen and his school, and many others).

• The Askey-Gasper inequality was the crucial fact needed in the proof of one of the most important open problems of the 20-th century, Louis de Branges’ proof of the Bieberbach conjecture.

Finally, let me mention Dick’s influence on myself.

• Dick was a great professional father-figure. He is the one who challenged me to prove George Andrews’ q-Dyson conjecture, that I did, in 1983 (published in 1985) in collaboration with Dave Bressoud. He is the one who challenged me to prove the G_{2} case of Ian Macdonald’s Constant Term Conjecture (that I did, also done independently by Laurent Habsieger), where I used the Dixon identity, that I learned from him and from Foata. Dick also challenged me to prove the G_{2}-dual case of the same conjecture, that I did using what I called the Stembridge-Stanton trick. That method was later used by another whiz, Frank Garvan, to prove the F_{4} case.

• Dick was an implicit, but very strong, influence in the development of so-called Wilf-Zeilberger algorithmic proof theory.

Thank you Dick!

Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8019, USA.

Email: DoronZeil at gmail dot com .