IN 1946, THE YEAR the ENIAC computer was built at Penn and Alexander Wilf joined the Irgun, a Roman Catholic nun had an epiphany of her own that was to have an important influence on Wilf's mathematical career. Sister Mary Celine Fasenmyer was raised in Central Pennsylvania and graduated from tiny Mercyhurst College after taking her vows. Because she was, as she later modestly put it, "always good in math," her order permitted her to pursue her Ph.D. at the University of Michigan during World War II under the supervision of the distinguished mathematician, Earl Rainville.
In her dissertation, Sister Celine showed that certain types of mathematical patterns known as recurrence relations could be satisfied in a mechanical, or algorithmic, way. Rainville recognized, as Wilf puts it, "that there was some very pretty mathematics going on," and presented his student's ideas to the world in two chapters of his book, Special Functions, published in 1960.
There Sister Celine's breakthrough remained for almost two decades, until, in 1978, Doron Zeilberger recognized that her technique could also be used to prove combinatorial identities. Wilf recalls first reading about Zeilberger's discovery in his office one quiet afternoon after grades had been submitted. "I remember feeling that I was about to connect to a parallel universe that had always existed but which until then had remained very well hidden, and I was about to find out what sort of creatures lived there."
As revolutionary as Zeilberger's application of Sister Celine's thesis may have been, it still had its rough edges. Wilf and Zeilberger had known each other casually since Zeilberger had sent Wilf a fan letter for another combinatorial paper Wilf had written, thanking him for restoring his faith in the beauty of mathematics. Wilf managed both to simplify Zeilberger's new technique and to devise a way to make it produce at least one, and sometimes more, entirely new and unanticipated identities in the process of proving the first. Their integrated discoveries became known as "WZ theory."
Zeilberger credits Wilf with making his idea aesthetically appealing. "Before Herb, it was correct, but it definitely was not pretty. His contribution made it beautiful." And this beauty is more than just skin deep. "It's not cosmetics," Zeilberger adds. "It leads to new breakthroughs. If it's not pretty, it's obscure."
Far from remaining obscure, their discoveries were set forth in the paper for which the two have been awarded this year's Steele Prize. They have since expanded upon it in a book published last year with Slovenian mathematician Marko Petkovsek, which they aptly titled A=B. Wilf and Zeilberger's paper met with enthusiastic reviews in the field. The eminent computer-scientist and mathematician Donald Knuth raved about it. "I fell in love with these procedures as soon as I learned them because they worked for me immediately," he wrote in the foreword to A=B. "The success rate was astonishing." So astonishing that Knuth abandoned two projects on which he had been working, regarding them as now obsolete.
In the midst of this success, however, Wilf did not forget Sister Celine. An enthusiastic amateur pilot, he flew to Erie where she lived in a Catholic retirement home and invited her to attend an upcoming mathematical conference in Boca Raton. Hearing no reply in the following weeks, Wilf flew down to Florida -- only to discover that Sister Celine had received a travel grant from the diocese and was in attendance.
When he introduced her from the audience, the 87-year-old nun slowly rose to her feet. She said she had only two remarks to make. First, she wanted to thank Professor Wilf for the invitation. And second, she said, casting a level gaze at the assemblage of distinguished mathematicians, "I want you all to know -- I really did that work."
"There wasn't a dry eye in the house," Wilf says.
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Copyright 1998 The Pennsylvania Gazette Last modified 5/15/98