# Leonard Carlitz

### Born: 26 December 1907 in Philadelphia, Pennsylvania, USA

Died: 17 September 1999 in Pittsburgh, Pennsylvania, USA

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**Leonard Carlitz**'s father, Michael Carlitz (born about 1877), was a Jewish immigrant from the Ukraine while his mother, Anna Schneyer (born about 1878), was a Jewish immigrant from Latvia. Michael, whose original name was Mordechai Karelitz, and Anna had both emigrated to the United States with their parents, the Carlitz family in 1891 and the Schneyer family in 1894. Michael Carlitz sold coffee in Philadelphia and, later, ran a tobacco store in the same city. Leonard, the middle child of is parents' three children, had two sisters, Roseline, born in Philadelphia around 1911, and Ruth, born in Philadelphia around 1903.

Leonard attended school in Philadelphia and his remarkable academic talents were clear at this time. These talents led to him being awarded a scholarship to study mathematics at the University of Pennsylvania in Philadelphia. He graduated with a A.B. in mathematics in 1927, completing his Master's degree in mathematics the following year. On completing his Master's degree, Carlitz remained at the University of Pennsylvania to study for his doctorate which was supervised by Howard H Mitchell (a student of Oswald Veblen). He became interested in Emil Artin's early work which was on quadratic number fields, in particular the analytic and arithmetic theory. In 1927 Artin made a major contribution to the theory of noncommutative rings, called hypercomplex numbers at this time. Inspired by Artin's work, Carlitz wrote his dissertation

*Galois fields of certain types*which led to the award of a doctorate in 1930 [8]:-

The main results of his doctoral dissertation were published in a 21-page paper in theOn his oral examination for the Ph.D. he was asked to sketch Gauss's proof of the law of quadratic reciprocity. Carlitz replied: "During his lifetime Gauss gave seven different proofs of this result. Which one would you like to see?"

*Transactions*of the American Mathematical Society in the year that his doctorate was awarded.

Immediately after the award of his doctorate, Carlitz received a National Research Council Fellowship which let him spend a year at the California Institute of Technology working with E T Bell. Returning to Philadelphia in the summer of 1931, he married Clara Skaler. They had two sons Michael Carlitz, born in 1939, and Robert D Carlitz born in 1945. Michael Carlitz became a IBM computer programmer while Robert became a University of Pittsburgh physics professor. After his marriage Carlitz and his wife headed for Cambridge in England where he spent the academic year 1931-32 as an International Research Fellow with G H Hardy [7]:-

When he returned to the United States he was offered a post at Duke University, in Durham, North Carolina, which he gladly accepted. In many ways he was fortunate to be offered this position since, although he had an outstanding record, this was the time of the Depression and jobs were extremely scarce. In a way he was lucky for two members of the Mathematics Department at Duke University, Joseph M Thomas and John H Roberts, both had connections to the University of Pennsylvania in Philadelphia and learnt first hand about Carlitz's talents. Roberts had been both an undergraduate and postgraduate student at the University of Pennsylvania and was awarded a Ph.D. in 1923. Duke University had been founded in 1924 but for the first years the Mathematics Department had been run by Robert Flowers, the University Vice-President, who had not been successful in appointing high quality researchers to the Department. This changed when Thomas and Roberts were appointed, in 1930 and 1932 respectively, who were then able to add Carlitz in 1932. Concerning Carlitz appointment to Duke University, Robert Durden writes in [1]:-This was the era when Hardy and Littlewood led one of the great centres of research in number theory, and Carlitz found the mathematical atmosphere there exhilarating. His work in additive number theory derives from that period.

The appointment of John Jay Gergen, who had been a student of Griffith Conrad Evans and Szolem Mandelbrojt, in 1936 strengthened the Duke Mathematics Department still further. Thomas, Roberts, Gergen and Carlitz taught almost all the mathematics graduate courses at Duke University until the 1950s. Carlitz remained on the Faculty at Duke from his appointment in 1932 until he retired in 1977. He was named James B Duke Professor of Mathematics in 1964.Carlitz ... had done both his undergraduate and doctoral work[at Pennsylvania]. His supervising professor there reported that Carlitz "so far excelled the other students here as to be literally in a class by himself." Only the "serious financial situation" at Pennsylvania, the professor explained, kept them from offering Carlitz a position there, and he was being considered at several major universities. From Oxford University, where Carlitz was doing postdoctoral work, came word that a paper published by him in the 'Oxford Quarterly Journal' had made an excellent impression and shown that he was "fully master of the technique of his trade." The Oxford don opined that Carlitz was no doubt "better equipped in the analytic theory of numbers than anyone else in America."

One of the important contributions that Duke University made to mathematics publishing was the founding of the

*Duke Mathematical Journal*in 1935. Carlitz was involved in the plans that were drawn up for the journal and he served on its editorial board from 1938 to 1973. The editorial board of this journal was set up in such a way that new members were appointed as necessary by the existing board. Carlitz was Managing Editor of the

*Duke Mathematical Journal*from 1945. However, in 1973 the administrators of the Duke mathematics department decided that they would decide on appointments to the board. Carlitz strongly opposed this and argued strongly that the board should keep its role in appointing new board members. When he was overruled, he resigned from the board. He also served on the editorial boards of other journals. these included

*Acta Arithmetica*,

*The American Mathematical Monthly*, and

*The Fibonacci Quarterly*.

Carlitz published 771 papers, supervised 45 doctoral students and 51 master's theses [8]:-

You can see a list of the 45 doctoral students with the year of their doctorate and the title of their thesis at THIS LINK.Carlitz was a kind and gentle man who cared deeply about his students and was admired by students and colleagues alike. Indeed, Carlitz was a mentor not only to his students but to a number of younger mathematicians.

His major mathematical contributions are to finite field theory, number theory, and combinatorics. But his publications extend beyond these areas to include algebraic geometry, commutative rings and algebras, finite differences, geometry, linear algebra, and special functions. David Hayes writes in [7]:-

Joel Brawley writes in [3] about his most remembered results (we have modified this quote by inserting the titles of the papers):-The importance of some of Carlitz's most profound papers was not appreciated until many years after they appeared in print. This unfortunate circumstance is sometimes attributed to the large number of his research papers. However, his choice of non-descriptive titles for many of his papers is a more likely explanation.

MathSciNet lists 237 papers with Carlitz's name in the title. Looking at the first few of these which occur in papers written in the last ten years we see: Carlitz's identity for the Bernoulli numbers; Al-Salam-Carlitz polynomials; Carlitz-typeOf all of his research, Carlitz will perhaps best be remembered for three items which bear his name. These are:(i)his development in[On certain functions connected with polynomials in a Galois Field(1935)]and[A class of polynomials(1938)]of what is today called the Carlitz module(see[D Goss, 'Basic Structures of Function Field Arithmetic'(Springer-Verlag, Berlin/New York,1996)]),(ii)a conjecture he made in1966which came to be known as the Carlitz Conjecture(see[D R Hayes, 'A geometric approach to permutation polynomials over a Finite Field', Duke Math. J.34(1967),293-305and also J V Brawley, 'Dedicated to Leonard Carlitz: The Man and his Work', Finite Fields Appl.1(1995),135-151]), and(iii)an estimate for exponential sums['Bounds for exponential sums'(1957)]that he derived with S Uchiyama, now known as the Carlitz-Uchiyama bound.

*q*-Euler polynomials; Carlitz's

*q*-Bernoulli numbers; Carlitz inversions and identities; Carlitz module analogues of Mersenne primes; Carlitz cyclotomic polynomials; Rademacher-Carlitz polynomials; Carlitz-Wan conjecture for permutation polynomials; Carlitz rank of permutations; Tricomi-Carlitz polynomials; Stieltjes-Carlitz polynomials; Carlitz's type q-tangent numbers; q-generating functions of the Carlitz; Carlitz-Srivastava polynomials; Carlitz's trilinear generating functions; Bernoulli-Carlitz numbers; Carlitz's q-operators; Dedekind-Carlitz polynomials; Carlitz algebras; Carlitz logarithms; Carlitz-Goss gamma function; and Carlitz compositions.

One has to ask; how anyone could publish 771 papers? Perhaps David Hayes provides at least a hint in [7]:-

As Joel Brawley writes in [2]:-In1953he published a record44papers. His most active decade was1960-69, when he averaged27papers per year. During the early1960s, when I was one of his graduate students, Carlitz had a National Science Foundation grant that paid for a half-time secretary. On more than one day I observed him reading a journal paper raising a question that he found of interest, that evening writing up a paper of his own answering the question, and having it typed and sent off to a journal the following day.

John Brillhart, a co-author of [4] and [5], is the editor-in-chief ofLeonard Carlitz is certainly one of the most prolific mathematical researchers of all time.

*Carlitz's Collected Works*and, given Carlitz's remarkable publication record, few editors of such volumes will have to handle a larger number of papers and the final result should contain around 6800 pages of research articles.

Hayes describes in [6] a graduate course that Carlitz gave at Duke University:-

One chapter of these chapters, Chapter 19, is published as an article inWhile a graduate student at Duke University in the early1960s, I was privileged to take Leonard Carlitz course "The arithmetic of polynomials." At the first class meeting, Leonard gave each student a detailed set of notes that served as a textbook for the course. These notes were neatly typed and reproduced by ditto machine, the leading-edge copying technique of that era. My own copy, still readable in faded ditto purple, occupies a special place on my office bookshelf. Of course the notes evolved over time. The version I have is277pages long and divided into29chapters.

*Finite Fields Appl*.

**1**(2) (1995), a volume specially dedicated to Carlitz. G L Mullen, the editor-in-chief, writes in the introduction to the part:-

Mullen, a doctoral student at the time, writes in [10] of meeting Carlitz:-With his permission, this article was extracted from classroom notes Carlitz wrote some forty to fifty years ago and shows that there is significance even in his unpublished research.

His hobbies and interests outside mathematics are given in [3]:-During the1973-74academic year, Professor Leonard Carlitz gave a colloquium in our department concerning permutation polynomials over finite fields. ... I vividly recall the tremendous enthusiasm that Professor Carlitz showed for various topics in finite field theory. In addition I recall how generous Professor Carlitz was with his ideas. Shortly after his return to Duke University, Professor Carlitz sent me a large package of reprints along with a number of very helpful comments and references.

Throughout his years at Duke University, the Carlitz family lived near the edge of the university campus at 1410 Markham Avenue, Durham. Clara Carlitz died in 1990 after their marriage which had lasted 57 years. Carlitz continued to live in his home on the edge of the campus until June 1999 when he was 92 years of age and his health was failing. He went into a nursing home in Pittsburgh, Pennsylvania at this time. This meant that he was close to his son Robert and Robert's family. On 15 September 1999 he contracted pneumonia and was taken to hospital. He died peacefully two days later.Throughout his life Carlitz was an avid reader and a great fan of classical music. Up until about a year prior to his death he still subscribed to the 'New York Times', and he was always very much interested in reading - especially biographies of the great scientists and mathematicians. Even in the several months prior to his death, Carlitz could readily identify the composers of the classical music pieces to which he was listening. His main form of exercise was walking, and for most of the45years he served as a member of the Duke faculty, twice a day he would walk the one-mile round trip between his university office and his home on the edge of the Duke campus. In his declining years he fell and broke his hip on three separate occasions, but each time with great determination relearned to walk - an indication of both his love for walking and his perseverance. He only stopped walking on doctor's orders about a year before his death.

Carlitz's contribution is summed up in [2]:-

Leonard Carlitz, a kind and gentle man with deep mathematical insights, has left a truly remarkable research record ...

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

**List of References** (10 books/articles)

**Mathematicians born in the same country**

**Additional Material in MacTutor**

**Other Web sites**