# Nathan Jacob Fine

### Born: 22 October 1916 in Philadelphia, USA

Died: 18 November 1994 in Deerfield Beach, Florida, USA

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**Nathan Fine**was known to his friends as Nat. He attended Temple University from where he was awarded a B.S. in 1936. He then moved to the University of Pennsylvania to study for his Master's degree which was awarded in 1939. Following this, he became a school teacher and taught at a junior high school from 1941 to 1942. His first university appointment came when he was appointed as Instructor in Mathematics at Cornell University for 1942, and his next was to the same position at Purdue University from 1942 to 1945. Of course these last positions were held during World War II and, while he held these posts, Fine undertook war work as a research mathematician at the Naval Ordnance Plant in Indianapolis during 1944-1945, and then he worked for the Operations Evaluations Group in Washington, D.C. during 1946-1947.

He undertook research at the University of Pennsylvania for a Ph.D. under the supervision of Antoni Zygmund and he obtained his doctorate in 1946 for his thesis

*On the Walsh Functions*. Zygmund, who had escaped with his wife and son from German controlled Poland to the USA in 1940, held a number of posts before he finally settled in Chicago. He was only at the University of Pennsylvania for a short while and Fine was lucky to have his doctoral studies supervised by such an outstanding mathematician. Fine was appointed to the position of Assistant Professor of Mathematics at the University of Pennsylvania in 1947 and in 1956 he was promoted to full professor. In 1963 he moved from the University of Pennsylvania to Pennsylvania State University, where he remained until his retirement in 1978.

Fine held a number of fellowships such as an NSF Postdoctoral Fellowship in 1953-54 and a Guggenheim Foundation Fellowships presented by the John Simon Guggenheim Memorial Foundation of New York City in 1958. The Mathematical Association of America had established The Earle Raymond Hedrick Lectures are named after the first president of the Association. The lecturer needed to be of known skill as an expositor of mathematics and be someone:-

Fine was honoured by being made Earle Raymond Hedrick Lecturer in 1966.... who will present a series of at most three lectures accessible to a large fraction of those who teach college mathematics.

As a mathematician Fine had wide interests publishing on many different topics including number theory, logic, combinatorics, group theory, linear algebra, partitions and functional and classical analysis. He is perhaps best known for his book

*Basic hypergeometric series and applications*published in the Mathematical Surveys and Monographs Series of the American Mathematical Society. The material which he presented in the Earle Raymond Hedrick Lectures twenty years earlier form the basis for the material in this text. Andrews has written an Introduction to the book in which he explains something of his own experiences of taking courses by Fine:-

Andrews writes more about Fine's lecturing style in [1]:-In1948, Nathan Fine published a note in the Proceedings of the National Academy of Sciences announcing several elegant and intriguing partition theorems. These results were marked both by their simplicity of statement and by the depth of their proof. Fine was at that time engaged in his own special development of q-hypergeometric series, and as the years passed he kept adding to his results and polishing his presentation. Several times, both at Penn and Penn State, he presented courses on this material. I took the course twice, first in1962-1963at Penn and then in1968-1969at Penn State. As a graduate student at Penn, I wrote my thesis on mock theta functions under Rademacher's direction. The material that Fine was lecturing about fit in perfectly with my thesis work and introduced me to many aspects of this extensive subject. The course was truly inspiring. As I look back at it, it is hard for me to decide whether the course material or Fine's exquisite presentation of it impressed me most.

We became somewhat diverted while looking at Fine's textMy fellow students and I viewed him as a mathematical juggernaut. He spoke deliberately and somewhat slowly(thank goodness). He smoked unfiltered Camel cigarettes which he Persian-inhaled throughout each lecture. I look back fondly on Nat, his beautiful mathematics, and his ever-present cigarettes. In each lecture, he was clearly having a whale of a good time, and so were we.

*Basic hypergeometric series and applications*when we began to look at Andrews' Introduction. Returning to the book itself, we quote from a review by David M Bressoud of the 124 page book:-

Many of Fine's papers were written jointly with other leading mathematicians. To quote just two examples, there isFor far too long, there has been a dearth of good references on basic hypergeometric series. The present book and Basic hypergeometric series by G Gasper and M Rahman have appeared in the past two years to greatly rectify this situation. It is a measure of the breadth of this field that after the respective first chapters there is virtually no overlap between these books.

Fine writes from the viewpoint of a number theorist, and his slim volume is rich with examples and results from the theory of partitions, the study of Ramanujan's mock theta functions, and modular equations. This is a very personal book, a distillation of those results in basic hypergeometric series which hold the most appeal to its author.

*The probability that a matrix be nilpotent*written jointly with Herstein and published in the

*Illinois Journal of Mathematics*in 1958, and

*Pairs of commuting matrices over a finite field*written jointly with Walter Feit and published in the

*Duke Mathematical Journal*in 1960. Fine was also interested in problem solving and contributed both problems and solutions to problems to several different journals. Coming out of this interest is, for example, his paper

*On rational triangles*in the

*American Mathematical Monthly*in 1976. A rational triangle is one with both rational sides and rational area. In this article Fine proved that here exist rational numbers

*a*and

*b*that are never sides of a rational triangle, and also there exists a rational triangle of any given rational area.

As to his interests, Andrews writes [1]:-

Let us end this brief biography by quoting a couplet Fine composed in honour of Erdős:-In addition to mathematics, he enjoyed a variety of games, including GO, chess, bridge, billiards, and backgammon. He was a life master at bridge and played duplicate bridge until two days before he died.

Chebychev said it, and I'll say it again,

There's always a prime betweenNand2N.

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

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**Mathematicians born in the same country**

**Honours awarded to Nathan Fine**

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1. | Hedrick lecturer | 1966 |

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