[From "Bulletin of the Institute of Combinatorics and its Applications", v. 44 [May 2005], p. 12]
Doron Zeilberger's mathematical career began in his early teens. He became enchanted by the beautiful world of mathematics, eventually receiving a PhD in mathematics from the Weizmann Institute of Science in 1976 under the direction of Harry Dym (an academic descendant of Courant and Hilbert). His early mathematical works were mainly in the theory of discrete analytic functions; in which he published several research papers in the late 1970's. In the early 1980's, he discovered some interesting results that connect partial difference equations and combinatorics. Since then, he has published over 140 research papers in combinatorics, number theory, and algorithmic proof theory.
Zeilberger's work on binomial coefficient identities, using his generalization of Sister Celine's technique, gave rise to what is now known as Wilf-Zeilberger (WZ) method which gives a unified approach for evaluating sums involving terms F(n,k) where F(n,k) is a hypergeometric term in both variables. Zeilberger [in collaboration with his student Moa Apagodu] has recently obtained a sharp upper bound for the orders of the recurrence relations generated by the Zeilberger and q-Zeilberger algorithms. This new algorithm has smaller program size complexity and provides an improved bound to those obtained using the older version.
The WZ method has added significance to the field of combinatorics. George Andrews, who is one of the leading experts in q-series wrote: "In my proof of Capparelli's conjecture, I was completely guided by the Wilf-Zeilberger method, even if I didn't use Doron's program [EKHAD] explicitly. I couldn't have produced my proof without knowing the principle behind the WZ-method".
Donald Knuth wrote, in his foreword to the book A=B by Petkovsek, Wilf, and Zeilberger: "Science is what we understand well enough to explain to a computer. Art is everything else. During the past several years, an important part of mathematics [binomial coefficient identities] has been transformed from an Art to a Science. No longer do we need brilliant insight to evaluate sums of binomial coefficients, we can now follow a mechanical procedure [guided by Zeilberger's program EKHAD] and discover the answer quite systematically."
Zeilberger's stellar achievements in combinatorics include proving the Alternating Sign Matrix Conjecture in 1996. His proof combines computer algebra and results from partition theory, symmetric functions, constant term identities, and difference operators. He has also proved Dyson's and [Andrews's] q-Dyson conjecture, the G2 and G2-dual cases of Macdonald's conjecture, and Julian West's conjecture on 2-stack-sortable permutations.
Recently, Zeilberger has written a 5-paper series on the so-called umbral transfer matrix method. He has blended the transfer matrix method of statistical physic with the umbral calculus to develop a method for counting difficult combinatorical structures, such as self-avoiding walks.
Zeilberger, a champion of using computers and algorithms to do mathematics quickly and efficiently, is in the forefront of current combinatorial research.