Nathan Fine Explained

Nathan Jacob Fine
Birth Date:	22 October 1916
Birth Place:	Philadelphia, USA
Death Place:	Deerfield Beach, Florida, USA
Citizenship:	American
Alma Mater:	University of Pennsylvania
Occupation:	Mathematician
Notable Works:	Basic Hypergeometric Series and Applications

Nathan Jacob Fine (22 October 1916 in Philadelphia – 18 November 1994 in Deerfield Beach, Florida) was an American mathematician who worked on basic hypergeometric series. He is best known for his lecture notes on the subject which for four decades served as an inspiration to experts in the field until they were finally published as a book. He solved the Jeep problem in 1946.

Nathan Fine retired in 1978 as a professor at Pennsylvania State University. Prior to that he had been on the faculties of the University of Pennsylvania and Cornell University. For a brief period (1946 - 1947) he also worked at the Operations Evaluation Group, affiliated with the Massachusetts Institute of Technology. Beside the book he published about 40 papers in several fields of mathematics. He is known for the Rogers-Fine identity.^[1]

Nathan Fine received his Ph.D. in 1946 from University of Pennsylvania, where he was a student of Antoni Zygmund. Fine was at the Institute for Advanced Study for the three academic years 1953–1954, 1958–1959, and 1959–1960.^[2] Fine's doctoral students include J. J. Price.

He wrote the book Basic Hypergeometric Series and Applications .^{[3] [4]}

Selected publications

• 10.1090/S0002-9904-1944-08085-3. The probability that a determinant be congruent to a (mod m). 1944. Fine. N. J.. 2. Niven. Ivan. Ivan M. Niven. Bulletin of the American Mathematical Society. 50. 2. 89–94. free.
• 88084. Fine. Nathan J.. 2. Some New Results on Partitions. Proceedings of the National Academy of Sciences of the United States of America. 1948. 34. 12. 616–618. 10.1073/pnas.34.12.616. 16588844. 1079179. 1948PNAS...34..616F. free.
• 10.1090/S0002-9947-1949-0032833-2. On the Walsh functions. 1949. Fine. N. J.. 2. Transactions of the American Mathematical Society. 65. 3. 372–414. free.
• 10.1090/S0002-9939-1950-0038397-5. Proof of a theorem of Jacobi. 1950. Fine. N. J.. 2. Proceedings of the American Mathematical Society. 1. 5. 666–667. free.
• 10.1090/S0002-9947-1950-0042535-2. The generalized Walsh functions. 1950. Fine. N. J.. 2. Transactions of the American Mathematical Society. 69. 66–77. free.
• 10.1090/S0002-9939-1951-0043194-1. Note on the Hurwitz zeta-function. 1951. Fine. N. J.. 2. Proceedings of the American Mathematical Society. 2. 3. 361–364. free.
• 10.1090/S0002-9939-1954-0064337-2. On the asymptotic distribution of certain sums. 1954. Fine. N. J.. 2. Proceedings of the American Mathematical Society. 5. 2. 243–252. free.
• 10.1073/pnas.41.8.588. Cesaro Summability of Walsh-Fourier Series. 1955. Fine. N. J.. 2. Proceedings of the National Academy of Sciences. 41. 8. 588–591. 16578449. 528139. 1955PNAS...41..588F. free.
• 10.1090/S0002-9947-1957-0091371-6. Fourier-Stieltjes series of Walsh functions. 1957. Fine. N. J.. 2. Transactions of the American Mathematical Society. 86. 246–255. free.
• 10.1090/S0002-9904-1960-10460-0. Extension of continuous functions in \beta N . 1960. Fine. N. J.. 2. Gillman. L.. Leonard Gillman. Bulletin of the American Mathematical Society. 66. 5. 376–382. free.
• 10.1090/S0002-9939-1962-0143172-5. Remote points in \beta R . 1962. Fine. N. J.. 2. Gillman. L.. Proceedings of the American Mathematical Society. 13. 29. free.
• 10.1090/S0002-9939-1965-0174934-9. Uniqueness theorems for periodic functions. 1965. Fine. N. J.. 2. Wilf. H. S.. Herbert Wilf. Proceedings of the American Mathematical Society. 16. 109–114. free.
• 10.1090/S0002-9904-1965-11381-7. The distribution of the sum of digits (mod p). 1965. Fine. N. J.. 2. Bulletin of the American Mathematical Society. 71. 4. 651–653. free.

External links

• "Nathan Fine 1916–1994" – biography article by George Andrews
• Fine's Equation – in MathWorld
• "On Fine's Partition Theorems, Dyson, Andrews and Missed Opportunities" – popular article by Igor Pak

Notes and References

1. Book: https://books.google.com/books?id=3bZQDwAAQBAJ&pg=PA71. Chapter 9. The Rogers-Fine identity. 71–79. Topics and Methods in q-Series. 9789813223387. Laughlin. James Mc. 22 September 2017.
2. Web site: Natham J. Fine. Institute for Advanced Study. 9 December 2019 .
3. Askey, Richard. Review: Nathan J. Fine, Basic hypergeometric series and applications. Bull. Amer. Math. Soc. (N.S.). 1989. 21. 1. 130–132. 10.1090/s0273-0979-1989-15789-3. free.
4. Book: Basic Hypergeometric Series and Applications. 9780821815243. Fine. Nathan Jacob. 1988.

   pbk edition

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