# Leonard Carlitz
**Leonard Carlitz** (December 26, 1907 – September 17, 1999) was an American mathematician at [Duke University](https://www.edgepedia.org/duke-university) who published more than 770 research papers across number theory, finite fields, and combinatorics, and whose name attaches to the Carlitz module, the Carlitz conjecture, and the Carlitz–Uchiyama bound.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) MathSciNet lists 237 papers with his name in the title, including Bernoulli–Carlitz numbers, Al-Salam–Carlitz polynomials, the Carlitz–Wan conjecture, the Carlitz rank of permutations, Dedekind–Carlitz polynomials, and the Carlitz–Goss gamma function.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/)
| Key fact | Detail |
|---|---|
| Life | Born Philadelphia, December 26, 1907; died September 17, 1999, aged 91[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) |
| Training | AB 1927, MA 1928, PhD 1930, University of Pennsylvania, under H. H. Mitchell; dissertation "Galois Fields of Certain Types"[3](https://fq.math.ca/Scanned/38-4/howard.pdf) |
| Output | 770 research papers by the AMS count, 771 by MacTutor and the Fibonacci Quarterly obituary; 44 papers in 1953 alone[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/) |
| Career | Duke University 1932–1977; James B. Duke Professor from 1964[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) |
| Students | 45 doctoral students (AMS Notices, Mathematics Genealogy Project) or 44 PhD plus 51 MS students (Howard); 119 genealogical descendants[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [3](https://fq.math.ca/Scanned/38-4/howard.pdf) • [4](https://www.mathgenealogy.org/id.php?id=20327) |
| Signature results | Carlitz module (1935/1938), Carlitz conjecture (1966, proved 1993), Carlitz–Uchiyama bound (1957)[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/) |
## Life and career
Carlitz grew up in Philadelphia and won a scholarship to the University of Pennsylvania, completing the AB in 1927, the MA in 1928, and the PhD in 1930, all in mathematics; his dissertation appeared in the 1930 *Transactions of the AMS* (volume 32, pp. 451–472).[3](https://fq.math.ca/Scanned/38-4/howard.pdf) He then held two postdoctoral years: 1930–31 as a National Research Council Scholar with [Eric Temple Bell](https://www.edgepedia.org/eric-temple-bell) at Caltech, and 1931–32 with G. H. Hardy at Cambridge.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) He spent 1935–36 at the [Institute for Advanced Study](https://www.edgepedia.org/institute-for-advanced-study).[3](https://fq.math.ca/Scanned/38-4/howard.pdf)
He joined Duke University in 1932 and remained until his retirement in 1977.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) He was involved in the early planning of the *Duke Mathematical Journal* (established 1935) and sat on its editorial board from 1938 to 1973, often as managing editor.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [3](https://fq.math.ca/Scanned/38-4/howard.pdf) In 1964 he was named James B. Duke Professor of Mathematics, the first member of his department to hold one of these distinguished chairs.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) The board service ended in conflict: in 1973 the Duke mathematics administrators decided to control appointments to the journal's editorial board, Carlitz argued that the board should keep that role, and when he was overruled he resigned.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/) He also served for many years on the editorial board of *The Fibonacci Quarterly*, in which he published 72 articles between 1963 and 1984, including 19 joint papers and 7 short notes.[3](https://fq.math.ca/Scanned/38-4/howard.pdf) He married Clara Skaler in 1931; they had two children, Michael (1939) and Robert (1945).[3](https://fq.math.ca/Scanned/38-4/howard.pdf)
## Major mathematical contributions
**q-analogs of classical numbers.** Around 1950 Carlitz introduced q-analogs of the [Bernoulli numbers](https://www.edgepedia.org/bernoulli-numbers), now called q-Bernoulli–Carlitz numbers; his 1948 paper "q-Bernoulli numbers and polynomials" remains among his most cited, with 175 zbMATH citations.[5](https://zbmath.org/authors/?q=ai:carlitz.leonard) • [7](https://www.numdam.org/item/JTNB_2017__29_2_347_0/) In function-field arithmetic he proved in 1935 that whenever s is a multiple of q−1,
\[ \zeta_{C}(s) = \frac{BC_{s}}{\Pi(s)} \cdot \tilde{\pi}^{\,s}, \]
an analog of [Euler's formula](https://www.edgepedia.org/eulers-formula) for even zeta values, where \( \tilde{\pi} \) is the Carlitz period, \( BC_{s} \) the Bernoulli–Carlitz numbers, and \( \Pi(s) \) the Carlitz factorial.[8](https://arxiv.org/html/2601.02162v1) Recent work continues to build on these objects: a 2017 paper represents the q-Bernoulli–Carlitz numbers as moments of orthogonal polynomials, yielding Hankel determinant factorizations and continued fractions for their generating series.[7](https://www.numdam.org/item/JTNB_2017__29_2_347_0/)
**Analysis in positive characteristic.** In a seminal paper Carlitz initiated analysis over fields of positive characteristic, introducing the appropriate notions of a factorial, an exponential, and a logarithm, together with a system of polynomials \( \{e_{i}\} \) now called the Carlitz polynomials.[9](https://ar5iv.labs.arxiv.org/html/math/0405543) Later work by Carlitz himself, [David Goss](https://www.edgepedia.org/david-goss), Dinesh Thakur, and others built analogs of the gamma, zeta, Bessel, and hypergeometric functions on this foundation.[9](https://ar5iv.labs.arxiv.org/html/math/0405543)
**Finite fields.** His 1957 paper "Bounds for exponential sums", with S. Uchiyama, established the Carlitz–Uchiyama bound on exponential sums.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/) In 1966 he conjectured that there are only finitely many [permutation polynomials](https://www.edgepedia.org/permutation-polynomial) of any given even degree over the totality of finite fields of odd order; this deep conjecture was proved in 1993 using the classification theorem for finite simple groups.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) The Carlitz–Wan conjecture, a related named item, is still cited in the literature.[2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/)
## The Carlitz module and finite fields
The construction now called the Carlitz module appeared in two papers, "On certain functions connected with polynomials in a Galois field" (1935) and "A class of polynomials" (1938).[2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/) Concretely, the datum of the degree-q polynomial \( \phi_{T}(x) = Tx + x^{q} \) together with the subsequent functions \( \phi_{a} \) forms the Carlitz module, the first instance of a Drinfeld module.[8](https://arxiv.org/html/2601.02162v1) Its significance is arithmetic: Carlitz's paper showed how to explicitly construct the abelian extensions of the function field \( \mathbb{F}_{q}(T) \), a solution of Hilbert's Twelfth Problem over \( \mathbb{F}_{q}(T) \), and Carlitz extensions of \( \mathbb{F}_{p}(T) \) are the function-field analogs of cyclotomic extensions of \( \mathbb{Q} \).[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [10](https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf)
**Neglect and rediscovery.** These papers were mostly forgotten for decades. David Hayes, a student of Carlitz, used the module in 1974 to give an explicit description of the maximal abelian extension of \( \mathbb{F}_{q}(T) \), analogous to the [Kronecker–Weber theorem](https://www.edgepedia.org/kronecker-weber-theorem), and [Vladimir Drinfeld](https://www.edgepedia.org/vladimir-drinfeld) was not aware of Carlitz's work when writing his own 1974 paper.[11](https://public.websites.umich.edu/~asnowden/seminar/2017/drinfeld/ODM.pdf) Carlitz's papers predate the work of Lubin and Tate by about 30 years; their neglect has been attributed partly to his over-productive output and bland paper titles, of which "A class of polynomials" is an example.[11](https://public.websites.umich.edu/~asnowden/seminar/2017/drinfeld/ODM.pdf)
**Connection to the Langlands program.** In Drinfeld's theory, rank 1 is where Drinfeld's framework meets Carlitz's constructions, while rank 2 mimics the classical theory of elliptic curves; Drinfeld's framework allowed [Laurent Lafforgue](https://www.edgepedia.org/laurent-lafforgue) to establish the Langlands correspondence for \( \mathrm{GL}_{r} \) over function fields in 2002.[8](https://arxiv.org/html/2601.02162v1) The Carlitz module's composition property obtained a far-reaching generalization in the theory of Drinfeld modules, the principal objects of function-field arithmetic.[9](https://ar5iv.labs.arxiv.org/html/math/0405543)
## Carlitz in combinatorics
Combinatorics is the largest single category in his collected works: 130 papers, ahead of finite fields (92) and Bernoulli, Euler, and [Stirling numbers](https://www.edgepedia.org/stirling-numbers) and polynomials (80), with separate categories for Eulerian numbers and polynomials (19) and Dedekind sums (21).[12](https://www.impan.pl/shop/en/publication/transaction/download/product/83168?download.pdf=) His most cited paper overall is "Degenerate Stirling, Bernoulli and Eulerian numbers" (1979), followed by the q-Bernoulli paper (1948, 175), "Eulerian numbers and polynomials" (1959, 94), and "Some orthogonal q-polynomials" with W. A. Al-Salam (1965, 94).[5](https://zbmath.org/authors/?q=ai:carlitz.leonard) By MSC classification, zbMATH assigns him 301 publications in number theory, 157 in combinatorics, and 38 in special functions.[5](https://zbmath.org/authors/?q=ai:carlitz.leonard) His positive-characteristic binomial relation has been used to develop an umbral calculus in the spirit of [Gian-Carlo Rota](https://www.edgepedia.org/gian-carlo-rota)'s school, the one documented point of contact with that tradition.[9](https://ar5iv.labs.arxiv.org/html/math/0405543)
## By the numbers
The publication counts differ between obituaries of similar standing: the AMS Notices memorial gives 770 research papers plus a final paper, drawn from his classroom notes, in the April 1995 issue of *Finite Fields and Their Applications*, while MacTutor and the Fibonacci Quarterly obituary credit 771 publications.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [2](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/) • [3](https://fq.math.ca/Scanned/38-4/howard.pdf) In 1953 he published a record 44 papers, and his most active decade was 1960–69, averaging 27 papers per year.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) zbMATH indexes 830 documents by Carlitz since 1930, including one book; OpenAlex records 830 articles, 9,818 citations, an h-index of 46, and an i10-index of 224.[5](https://zbmath.org/authors/?q=ai:carlitz.leonard) • [6](https://openalex.org/authors/a5072073027) The student counts also differ slightly: 45 doctoral students per the AMS Notices and the Mathematics Genealogy Project (which lists 119 descendants), against 44 PhD students plus 51 MS students in Howard's obituary.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf) • [3](https://fq.math.ca/Scanned/38-4/howard.pdf) • [4](https://www.mathgenealogy.org/id.php?id=20327) Most of the dissertations involved finite fields.[1](https://www.ams.org/notices/200111/comm-carlitz.pdf)
His papers are preserved in print: the *Collected Papers of Leonard Carlitz* were organized into 15 categories with editorial help from [Waleed Al-Salam](https://www.edgepedia.org/waleed-al-salam), Joel Brawley, John Brillhart (main editor), [Henry Gould](https://www.edgepedia.org/henry-w-gould), David Hayes, Basil Gordon, Theresa Vaughan, and Albert Leon Whiteman, the editors having to reconstruct an accurate bibliography from disorderly and incomplete records.[12](https://www.impan.pl/shop/en/publication/transaction/download/product/83168?download.pdf=) A complete bibliography by Brawley, Brillhart, and Gould appeared in *Acta Arithmetica*, volume 152 (2012), no. 4, pp. 361–405.[13](https://geodesic.mathdoc.fr/articles/10.4064/aa152-4-3/)
## Recent developments
Carlitz-module research remains active. A February 2025 preprint studies the formal Carlitz module over finite extensions of \( \mathbb{F}_{q}(\theta) \) and monic irreducible polynomials in \( \mathbb{F}_{q}[\theta] \), building on the work of Greg Anderson and Dinesh Thakur on log-algebraicity.[14](https://arxiv.org/abs/2502.08159) A 2026 survey notes that the algorithmic aspects of Drinfeld modules, despite their remarkable impact, are a subject of very recent developments, with cited works from 2018 through 2026.[8](https://arxiv.org/html/2601.02162v1) Two further 2026 preprints extend the theory: one develops a comprehensive parallel to Carlitz module theory for infinite places of degree greater than 1, filling a longstanding gap, and shows a critical distinction from Carlitz theory in that the standard Anderson generating function residue formula fails there due to [Galois group](https://www.edgepedia.org/galois-group) action; another gives explicit formulas for the exponential and logarithm of the n-th tensor power of the Carlitz module (introduced by Anderson and Thakur in 1990) and uses them to prove transcendence results for log-type hypergeometric functions over function fields.[15](https://ar5iv.labs.arxiv.org/html/2605.07484) • [16](https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1278/)
## References
1. [Leonard Carlitz (1907–1999), Notices of the AMS, Vol. 48, No. 11](https://www.ams.org/notices/200111/comm-carlitz.pdf)
2. [Leonard Carlitz (1907–1999), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Carlitz/)
3. [F. T. Howard, In Memoriam—Leonard Carlitz, Fibonacci Quarterly 38:4](https://fq.math.ca/Scanned/38-4/howard.pdf)
4. [Leonard Carlitz, The Mathematics Genealogy Project](https://www.mathgenealogy.org/id.php?id=20327)
5. [zbMATH author profile: Leonard Carlitz](https://zbmath.org/authors/?q=ai:carlitz.leonard)
6. [L. Carlitz, OpenAlex](https://openalex.org/authors/a5072073027)
7. [Nombres de q-Bernoulli–Carlitz et fractions continues, J. Théor. Nombres Bordeaux, 2017](https://www.numdam.org/item/JTNB_2017__29_2_347_0/)
8. [A computational approach to Drinfeld modules, arXiv](https://arxiv.org/html/2601.02162v1)
9. [Umbral Calculus in Positive Characteristic, arXiv math/0405543](https://ar5iv.labs.arxiv.org/html/math/0405543)
10. [Carlitz Extensions, Keith Conrad expository notes](https://kconrad.math.uconn.edu/blurbs/gradnumthy/carlitz.pdf)
11. [Drinfeld modules and the Carlitz module, Snowden seminar notes](https://public.websites.umich.edu/~asnowden/seminar/2017/drinfeld/ODM.pdf)
12. [Collected Papers of Leonard Carlitz, editorial introduction](https://www.impan.pl/shop/en/publication/transaction/download/product/83168?download.pdf=)
13. [Brawley, Brillhart, Gould: Recollections of Leonard Carlitz, Acta Arithmetica 152 (2012)](https://geodesic.mathdoc.fr/articles/10.4064/aa152-4-3/)
14. [On the formal Carlitz module, arXiv (2025)](https://arxiv.org/abs/2502.08159)
15. [Anderson generating function of rank-one Drinfeld modules over rational function fields, arXiv (2026)](https://ar5iv.labs.arxiv.org/html/2605.07484)
16. [Explicit formulas for the exponential and logarithm of the Carlitz–Tate twist, JTNB](https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1278/)
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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Mathematicians and statisticians › Number theorists › Algebraic number theorists*
*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026*
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