# Herbert Saul Wilf

### Born: 13 June 1931 in Philadelphia, Pennsylvania, USA

Died: 7 January 2012 in Wynnewood, Pennsylvania, USA

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**Herbert Wilf**'s parents, Alexander and Bertha Wilf, were Jewish. Alexander Wilf (1904-1980) had a profitable business in Philadelphia selling rugs, carpets and appliances. Herbert had a younger sister Gladys, born in 1939. He attended primary school in Philadelphia, and then continued his education in his home town at the Central High School. Even as a child, Herb (as he was known to his friends and colleagues) was fascinated by mathematics and he would read mathematics books in his bed after he had been put down to sleep by his parents. It was while he was at the Central High School that his father made a decision which had a huge impact on the family.

Alexander Wilf, known as Alex, had right-wing Zionist views and was a passionate supporter of the idea of creating a Jewish state in the middle east. He founded the local branch of the Committee for a Jewish Army in 1941. Three years later he decided to devote his life to that cause, sold his home in Wynnefield together with the family business, and moved to New York to become the executive director of the American League for a Free Palestine. Herb continued to study at the Central High School in Philadelphia and lived with some of his relatives. He felt, in his own words, "deserted by his family". Alexander Wilf was involved in smuggling refugees and arms into Palestine, using ships in which he had invested most of his wealth. He had invested money in the

*Abril*, which sailed to Palestine in March 1947 with over 600 immigrants and crew. Alex Wilf also put money into the

*Altalena*which sailed for Tel Aviv in June 1948 with 940 fighters and a large quantity of arms and ammunition. However, there was great differences of opinion among the leaders in Israel about accepting this cargo when a ceasefire was in operation. Many were killed in the resulting conflict when the ship was shelled. The Altalena Affair exposed deep rifts between the main political factions in Israel.

Herb Wilf completed his schooling in 1948 and, following his father's advice, applied to study at the Massachusetts Institute of Technology. He said [1]:-

Before he had begun his studies, however, with the state of Israel now set up, Alexander Wilf decided to take his family to Israel and start up a newspaper. He pressed Herb to come to Israel with the rest of the family, but he resisted [1]:-My father said, 'Why don't you go to MIT and be a scientist, because that's what you're good at. I had so much faith in my father that I applied just to MIT - nothing else - and I got in! I thought it was that way for everybody.

Wilf was awarded a B.S. in mathematics from the Massachusetts Institute of Technology in 1952. However, he had already made important contacts [8]:-I can't quite figure myself out at the point, but I was very, very resistant. I was very strongly on a professional track. I really wanted to be a scientist ... and I was just as stubborn as my father was.

After graduating from MIT, Wilf went to Columbia University where he studied for a Ph.D. supervised by Herbert Ellis Robbins, who was the Professor of Mathematical Statistics. However, although Robbins was his official advisor, in fact the inspiration behind his doctoral thesis came from Gerald Goertzel, Professor of Physics at New York University [8]:-In1951, I was a junior at MIT, and one weekend I was in New York City and I wanted to play some duplicate bridge. I went to the headquarters of the American Contract Bridge League, in the west50's in Manhattan, to find a list of tournaments that I might play in. It was a Saturday, though, and their office was closed. Next door I saw an office marked "Nuclear Development Associates", a name that intrigued me because I was then interested in nuclear physics, so I walked in and was very cordially received. In fact I was offered a summer job on the spot.

While he was undertaking graduate studies at Columbia University, Wilf married Ruth Tumen; they had a daughter Susan (who became a Chinese scholar), and two sons David (who became a lawyer) and Peter (who became a palaeontologist). With a wife and young family to support, Wilf needed to earn money and took on a number of jobs such as designing jet engines with the Fairchild Engine Division in 1954 and as head of the Computing Section of Nuclear Development Associates 1955-59. While he was at graduate school, his parents and sister returned from Israel after the newspaper that his father had set up there failed. Wilf was awarded his Ph.D. in 1958 and, in the following year, he was appointed as an Assistant Professor of Mathematics at the University of Illinois. After three years in this position, Wilf was appointed as Associate Professor of Mathematics at the University of Pennsylvania. He worked for the University of Pennsylvania for the rest of his career being promoted to Professor of Mathematics in 1965. One of the earliest endowed chairs at the University of Pennsylvania was the Thomas A Scott Professorship established by the Trustees in June 1881. In January 1998 Wilf was named Thomas A Scott Professor.In1952, when I met Jerry, it was, for me anyway, the dawn of the computer era, and I was doing as much programming as I could because I enjoyed it. Jerry taught me about programming and about computers.[The]technical work for my thesis was done with the inspiration and guidance of Jerry Goertzel. The title of the thesis was "The transmission of neutrons in multilayered slab geometry." It solved the transport equation in multilayered geometry by regarding each homogeneous layer as a little black box with prescribed inputs and outputs(which point of view was Jerry's hallmark), and it wired them together by representing each by a matrix. Then to get through the various different layers, one simply multiplied the matrices together. My name was on the thesis, but it was Jerry's philosophy that made it happen, along with his specific advice on numerous technical problems. He was quite definitely my mentor, though that word never occurred to me.

Even before the award of his doctorate, Wilf had written a remarkable range of papers: (with M Kalos)

*Monte Carlo solves reactor problems*(1957);

*An open formula for the numerical integration of first order differential equations*(1957);

*An open formula for the numerical integration of first order differential equations. II*(1958);

*Curve-fitting matrices*(1958);

*A stability criterion for numerical integration*(1959); and

*Matrix inversion by the annihilation of rank*(1959). Also early in his career he began publishing influential texts. The first,

*Mathematical Methods for Digital Computers*(1960), written with A Ralston, came out of the work that he had done for Nuclear Development Associates while he was still a research student. His next book,

*Mathematics for the Physical Sciences*(1962) was intended for:-

The book contained ideas that he had studied while a research student. He wrote [8]:-... physicists, engineers and other natural scientists in their first or second year of graduate study.

However, Wilf is best known for his remarkable contributions to combinatorics [5]:-My name was on the book, but Jerry Goertzel's ideas and inspiration were the core of the presentation of orthogonal polynomials and Gauss Quadrature via tridiagonal matrices and their spectra.

Wilf's most memorable contribution in this area is now known as 'WZ theory'. This is work which he undertook jointly with Doron Zeilberger resulting in their joint paperIn the1960's, Wilf became interested in the newly developing field of combinatorial analysis. He wrote fundamental research papers, forming the foundation of today's work in discrete mathematics with its applications to computer algorithms and its close interconnections with the mathematical fields of algebra and probability theory.

*Rational functions certify combinatorial identities*(1990). For this work Wilf and Zeilberger were jointly awarded the Leroy P Steele Prize by the American Mathematical Society at the January 1998 meeting of the Society in Baltimore. The citation for the award reads:-

Wilf's response to receiving the Prize explains how the joint work came about:-New mathematical ideas can have an impact on experts in a field, on people outside the field, and on how the field develops after the idea has been introduced. The remarkably simple idea of the work of Wilf and Zeilberger has already changed a part of mathematics for the experts, for the high-level users outside the area, and the area itself.

We must also mention Wilf's contributions to mathematics teaching. The quality of this is seen from the wards that he received: the Christian and Mary Lindback Award for excellence of undergraduate teaching from the University of Pennsylvania (1973); the Award for Distinguished College or University Teaching of Mathematics, Eastern Pennsylvania and Delaware Section of the Mathematical Association of America (1995); and the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics from the Mathematical Association of America in January, 1996. This last mentioned award was instituted:-I am deeply honoured to receive the Leroy P Steele Prize. I might say that doing this research was its own reward - but it's very nice to have this one too! My thanks to the Selection Committee and to the American Mathematical Society. Each semester, after my final grades have been turned in and all is quiet, it is my habit to leave the light off in my office, leave the door closed, and sit by the window catching up on reading the stack of preprints and reprints that have arrived during the semester. That year, one of the preprints was by Zeilberger, and it was a21st century proof of one of the major hypergeometric identities, found by computer, or more precisely, found by Zeilberger using his computer. I looked at it for a while and it slowly dawned on me that his recurrence relation would assume a self dual form if we renormalize the summation by dividing first by the right hand side. After that normalization, the basic "WZ" equation F(n+1, k)- F(n, k)= G(n, k+1)- G(n, k)appeared in the room, and its self-dual symmetrical form was very compelling. I remember feeling that I was about to connect to a parallel universe that had always existed but which had until then remained well hidden, and I was about to find out what sorts of creatures lived there. I also learned that such results emerge only after the efforts of many people have been exerted, in this case, of Sister Mary Celine Fasenmyer, Bill Gosper, Doron Zeilberger and others.

Let us quote from Wilf to get a feeling for his views on mathematical education:-... to honour college or university teachers who have been widely recognized as extraordinarily successful and whose teaching effectiveness has been shown to have had influence beyond their own institutions.

Let us look briefly at the books that Wilf has published, in addition toA lot of attention is paid in this country to the programmatic aspects of education; that is to say, to curricula. Shall we teach this or that or the other thing, and if so in what grade and by what textbook? I think that's mostly irrelevant. Teaching is mostly a chemical activity that goes on between Human A and Human B. Forget the curriculum. ... The cost to society of regimenting kids so they all listen to their teacher and go home and do homework for four hours is tremendous. I do not want this society to take that course. I want my kids to be free and untrammelled to screw up their mathematics and flunk all those standardized tests. ... The biggest single problem we have to block understanding in mathematics is the fact that instructors don't push students hard enough to force them to verbalize it. Yes, mathematics is a universal language, but you've got to speak it. It's a formal language. You can't use informal conversation. It won't work. ... what is at the highest premium is the ability of students to wrap complete English sentences around their mathematical thoughts.

*Mathematics for the Physical Sciences*(1962) which we mentioned above. In 1970 he published

*Finite sections of some classical inequalities*which studied spectral theory of finite sections of infinite-dimensional operators. A E Heins writes in a review:-

In 1975, in collaboration with Albert Nijenhuis, he publishedThe author is to be commended for having given us a connected and clear account of the interesting work developed mainly by M Kac, G Szegő, H Widom, and N de Bruijn during the last two decades.

*Combinatorial algorithms*. This book presented a collection of programs, written in FORTRAN, for various combinatorial algorithms. Wilf's 1986 book

*Algorithms and complexity*was reviewed by Charles Colbourn:-

In 1990 he published a book with a rather non-standard title, namelyThis is an introductory textbook on the design and analysis of algorithms. The main strengths of the book are its very clear writing style and strong emphasis on motivation.

*generatingfunctionology*. Philip Hanlon begins a review by writing:-

Finally, there is Wilf's remarkable book again with a rather strange title, this timeAre there such words as "generatingfunctionology", "generatingfunctionological", and "generatingfunctionologists"? In this book the author asserts that there are, but I have my doubts. This book contains an introduction to the topic of generating functions that is suitable for an advanced undergraduate. Ordinarily, generating functions are covered in a more comprehensive course on combinatorics and graph theory. My experience has been that students have difficulty understanding generating functions. This is not surprising - they seem to be functions of x yet we pay more attention to their coefficients than their values. They solve complicated enumeration problems in mysterious, magical ways. The problem here is neither the students nor the topic. We just do not take the time in a typical undergraduate combinatorics course to give a slow, careful treatment of generating functions. That is exactly what the author does in this book.

*A = B*. This book describes the work by Wilf and Zeilberger which led to them receiving the Leroy P Steele Prize by the American Mathematical Society and was jointly authored by Wilf, Marko Petkovsek and Doron Zeilberger. The book contains a Foreword written by Donald Knuth who writes:-

Finally we mention that Wilf founded two journals,Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coefficients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically.

*The Journal of Algorithms*in 1980 with co-founder Donald Knuth, and

*The Electronic Journal of Combinatorics*in 1994 with co-founder Neil Calkin. He also served as the editor-in-chief of the

*American Mathematical Monthly*from 1987 to 1992 and on the editorial board of

*Discrete Mathematics and Theoretical Computer Science*from 1999-2011.

Outside of mathematics Wilf had a passion for flying:-

Wilf died of a progressive neuromuscular disease. The traditional days of mourning, shivah, were observed at his home in Wynnewood and a memorial service was held on 15 January 2012 at Temple Beth Hillel-Beth El in Wynnewood. The Herbert S Wilf Award, recognising outstanding student achievement, is being set up at the Department of Mathematics, The University of Pennsylvania.From about1982until2002I flew a single engine airplane for a hobby. It gave my wife and me a great deal of pleasure, as well as a few hair raising moments, and we saw a great deal of North America as a result.

**Article by:** *J J O'Connor* and *E F Robertson*

**List of References** (9 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

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