# George W Eyre Andrews

### Quick Info

Salem, Oregon, USA

**George E Andrews**is an American mathematician who worked in special functions, number theory, analysis and combinatorics

### Biography

**George E Andrews**' parents were Raymond Leslie Andrews and Ravena Pearl Eyre. He grew up on a farm outside Salem. He liked detective stories when he was at junior high school and realised at that time that he enjoyed working things out. He thought that he might achieve this by entering the legal profession, but when he was at high school his ideas changed somewhat and he decided that he would study engineering. It was with this in mind that he began his undergraduate studies at Oregon State University in Corvallis in 1956. However, his mathematics lecturer soon persuaded him that he should forget his idea of taking engineering as his major subject and choose mathematics instead. Andrews said [1]:-

I went into mathematics and never looked back. I was struck by the beauty and the appeal of mathematics. It just captivated me.Andrews graduated from Oregon State University with both a B.S. and an M.S. in June 1960. Later that summer, on 2 September, he married Joy Margaret Brown; they had three children Amy Beth, Katherine Yvonne, and Derek George. He then spent a year at the University of Cambridge in England on a Fulbright Scholarship. After returning to the United States in 1961, Andrews began his graduate studies at the University of Pennsylvania in Philadelphia. He already knew that he wanted to undertake research in number theory but at the time he began his course it was prime numbers which fascinated him. He took a graduate course on number theory given by Hans Rademacher and there he met the notion of a partition. Andrews had come across these ideas before, since Joy, before they were married, had given him a book which contained G H Hardy's

*A Mathematician's Apology*and in that he had encountered the partition formula that Hardy and Ramanujan had discovered in 1916 for the number of different ways an integer can be expressed as a sum of natural numbers. Andrews was fascinated and asked Rademacher if he would become his thesis advisor. The topic suggested by Rademacher was Ramanujan's mock theta functions and indeed his research led to his thesis

*On the Theorems of Watson and Dragonette for Ramanujan's Mock Theta Functions*. Andrews had published three papers by the time he had completed his thesis work:

*An asymptotic expression for the number of solutions of a general class of Diophantine equations*(1961);

*A lower bound for the volume of strictly convex bodies with many boundary lattice points*(1963); and

*On estimates in number theory*(1963). This last paper, in the

*American Mathematical Monthly*, gave a method for finding an upper bound for the number of solutions of a Diophantine equation of the form $y = f (x)$.

Andrews was awarded his doctorate in 1964 from the University of Pennsylvania and was offered the position of assistant professor at Pennsylvania State University. His list of publications continued to grow with the paper

*A simple proof of Jacobi's triple product identity*(1965) appearing before three papers were published in the following year based on the work of his doctoral thesis on mock theta functions and partitions. In 1967 Andrews was promoted to associate professor and then to full professor in 1970. He continued his career at Pennsylvania State University, being named Evan Pugh Professor in 1981.

The year 1970 saw the publication of Andrews' first book

*Number theory*. C Ryavec writes:-

This book is highly suitable for a course in elementary number theory. The material is presented clearly, and it is supplemented by a balanced selection of exercises, so that the interest of both the merely competent and of the more able undergraduate is maintained.In 1970-71 Andrews was a visiting professor at Massachusetts Institute of Technology, then in 1975 at the University of Erlangen in Germany. Back in the United States he went to the University of Wisconsin where he was a visiting professor during 1975-76 working mainly with Richard Askey. Having received an invitation to speak at a conference in Strasbourg, France, he went there from Wisconsin. After the conference ended Andrews visited the University of Cambridge in England before returning to Wisconsin. In Cambridge Andrews asked to be allowed to work on the papers left by G N Watson on his death, which had been donated to the University Library and continued to be housed there. What Andrews found in an old box astounded him, for it contained around 100 loose pages in what he immediately recognised as Ramanujan's handwriting. It did not take Andrews long to recognise that the formulas on the sheets concerned mock theta functions. Indeed there was little else but formulas on the sheets with hardly any connecting words. Askey has said [1]:-

There were certain identities that George recognized as mock theta functions. No one else would have spotted them instantly. George has done many things, but this is what will make the history books.About the same time as Andrews was making his major discovery of what is now called 'Ramanujan's lost notebook', his second book

*The theory of partitions*was published. Carlitz writes:-

... as the author points out, there are almost no books devoted entirely to partitions. The combinatorial and formal power series aspects of the subject have usually been treated in books on elementary number theory or combinatorial analysis. Asymptotic problems concerning partitions have been considered in books on analytic or additive number theory. Applications of partitions in various branches of mathematics and statistics, the author points out, may make use of both combinatorial and asymptotic methods. ...Andrews, on his web page, describes his research interests as follows:-

The book is very well written. Despite the large amount of ground covered it is quite readable. It is an excellent introduction to a fascinating subject.

My research centres on the theory of partitions and related areas. I have a long-term interest in the work of Ramanujan, the Indian genius, whose last notebook I unearthed in the Trinity College Library at Cambridge in 1976. I am collaborating with Bruce Berndt on a multi-volume study of this 'Lost Notebook.' The first volume of this study appeared in June 2005. In addition, I have written more than 250 scientific papers, and several books on number theory and the theory of partitions, as well as edited the collected papers of Percy A MacMahon.As Andrews says, the first volume of Ramanujan's lost notebook, co-authored with Bruce C Berndt, was published in 2005. It is the first of what the authors expect to be four volumes. Andrew V Sills writes:-

The mathematics community owes a huge debt of gratitude to Andrews and Berndt for undertaking the monumental task of producing a coherent presentation along with complete proofs of the chaotically written mathematical thoughts of Ramanujan during the last year of his life. Some 85 years after his death, beautiful "new" and useful results of Ramanujan continue to be brought to light.Among Andrews' other books we mention

*Special functions*written jointly with Richard Askey and Ranjan Roy and published in 1999. Bruce C Berndt, in a review of the book, points to many remarkable features. We extract a few:-

The book genuinely reflects the author's vast accumulated insights. Most notably, the authors demonstrate a superb familiarity with the historical roots of their subject. ...It is impossible to give any sort of overview of the totality of Andrews' contributions given that he has, up to 2006, 275 items in his publications list. Although he has continued on the faculty at Pennsylvania State University, Andrews has spent around 12 further spells as visiting professor at universities in the United States, Canada, Mexico, Australia, France, South Africa, Austria and England.

"Special functions" will certainly emerge as the chief textbook and reference on special functions for the next several years. Indeed, the historical insights and copious problems make it an excellent textbook for a beginning graduate course. This book joins F W J Oliver's 'Asymptotics and special functions', first published in 1974, as the only general books on special functions during the past three decades that belong "in the Hobbs class", to quote G H Hardy.

He has served on the editorial boards of many journals:

*Discrete Mathematics*, the

*Journal of Combinatorial Theory (A); The Ramanujan Journal; Integers: The Electronic Journal of Combinatorial Number Theory*; the

*Proceedings of the Jangjeon Mathematical Society; Advanced Studies in Contemporary Mathematics*; the

*Acharya Nagarjuna International Journal of Mathematics and Information Technology*; and the

*International Journal of Number Theory*.

Andrews has received, and continues to receive, many honours for his outstanding contributions. For example he has been an invited one-hour speaker at four Joint American Mathematical Society - Mathematical Association of America meetings (1972, 1977, 1993, 1994), and he was the Mathematical Association of America's Hedrick Lecturer in 1980, and the J S Frame Lecturer in 1993. He has received honorary degrees from the University of Parma (1998), the University of Florida (2002), and the University of Waterloo (2004). He was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences in 2003. He has received the Allegheny Region Distinguished Teaching Award from the Mathematical Association of America and the Centennial Award from the Department of Mathematics, the University of Pennsylvania:-

... in recognition of contributions to pure mathematics and [...] mathematical education ...in October 1999.

As to his current (September 2006) research interests, Andrews writes:-

Currently I am reviving MacMahon's "Partition Analysis", collaborating on further applications of partitions to statistical mechanics and computer science, and completing my study of the relationship between Ramanujan's enigmatic identities and quadratic forms.

### References (show)

- R Nuzzo, Biography of George E Andrews,
*Proc. Natl. Acad. Sci. USA***102**(13) (2005), 4663-4665.

### Additional Resources (show)

Other websites about George E Andrews:

### Honours (show)

Honours awarded to George E Andrews

### Cross-references (show)

Written by J J O'Connor and E F Robertson

Last Update November 2006

Last Update November 2006