Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Continued fractions and modular functions


Author: W. Duke
Journal: Bull. Amer. Math. Soc. 42 (2005), 137-162
MSC (2000): Primary 11Fxx, 11Gxx
DOI: https://doi.org/10.1090/S0273-0979-05-01047-5
Published electronically: January 25, 2005
MathSciNet review: 2133308
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [An1] George E. Andrews, An introduction to Ramanujan’s “lost” notebook, Amer. Math. Monthly 86 (1979), no. 2, 89–108. MR 520571, https://doi.org/10.2307/2321943
  • [An2] George E. Andrews, Ramanujan’s “lost” notebook. I. Partial 𝜃-functions, Adv. in Math. 41 (1981), no. 2, 137–172. MR 625891, https://doi.org/10.1016/0001-8708(81)90013-X
    George E. Andrews, Ramanujan’s “lost” notebook. II. 𝜗-function expansions, Adv. in Math. 41 (1981), no. 2, 173–185. MR 625892, https://doi.org/10.1016/0001-8708(81)90014-1
    George E. Andrews, Ramunujan’s “lost” notebook. III. The Rogers-Ramanujan continued fraction, Adv. in Math. 41 (1981), no. 2, 186–208. MR 625893, https://doi.org/10.1016/0001-8708(81)90015-3
  • [Art] Emil Artin, Galois theory, 2nd ed., Dover Publications, Inc., Mineola, NY, 1998. Edited and with a supplemental chapter by Arthur N. Milgram. MR 1616156
  • [Ask] Richard Askey, Orthogonal polynomials and theta functions, Theta functions—Bowdoin 1987, Part 2 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 299–321. MR 1013179
  • [BC] S. Barnard and J. M. Child, Advanced Algebra, Macmillan and Co., Ltd., London, 1939. MR 0001185
  • [Be] Bruce C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR 781125
    Bruce C. Berndt, Ramanujan’s notebooks. Part V, Springer-Verlag, New York, 1998. MR 1486573
  • [BCZ] Bruce C. Berndt, Heng Huat Chan, and Liang-Cheng Zhang, Explicit evaluations of the Rogers-Ramanujan continued fraction, J. Reine Angew. Math. 480 (1996), 141–159. MR 1420561
  • [Ber] W. E. H. Berwick, Modular invariants expressible in terms of quadratic and cubic irrationalities. Proc. London Math. Soc. (2) 28, 53-69 (1928).
  • [Bir] B. J. Birch, Weber’s class invariants, Mathematika 16 (1969), 283–294. MR 0262206, https://doi.org/10.1112/S0025579300008251
  • [Bor] Richard E. Borcherds, What is Moonshine?, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 607–615. MR 1660657
  • [BCHIS] A. Borel, S. Chowla, C. S. Herz, K. Iwasawa, and J.-P. Serre, Seminar on complex multiplication, Seminar held at the Institute for Advanced Study, Princeton, N.J., 1957-58. Lecture Notes in Mathematics, No. 21, Springer-Verlag, Berlin-New York, 1966. MR 0201394
  • [Bra] R. Brauer, Galois Theory, 1957-1958, Harvard Lecture Notes.
  • [Cox] David A. Cox, Primes of the form 𝑥²+𝑛𝑦², A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989. Fermat, class field theory and complex multiplication. MR 1028322
  • [Dar] H. B. C. Darling, Proofs of certain identities and congruences enunciated by S. Ramanujan. Proc. Lond. M. S. (2) 19, 350-372 (1921).
  • [Deu] M. Deuring, Die Klassenkörper der komplexen Multiplikation, Enzyklopädie der mathematischen Wissenschaften: Mit Einschluss ihrer Anwendungen, Band I 2, Heft 10, Teil II (Article I 2, vol. 23, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1958 (German). MR 0167481
  • [Dic] L. E. Dickson, Modern algebraic theories. Sanborn, New York, 1926.
  • [Eis] G. Eisenstein, Theorema. J. für die reine und angew. Math. 29, 96-97 (1845) [ #26 in Mathematische Werke. Band I. 289-290, Chelsea Publishing Co., New York, 1975].
  • [Elk] Noam D. Elkies, The Klein quartic in number theory, The eightfold way, Math. Sci. Res. Inst. Publ., vol. 35, Cambridge Univ. Press, Cambridge, 1999, pp. 51–101. MR 1722413
  • [FK] Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, Graduate Studies in Mathematics, vol. 37, American Mathematical Society, Providence, RI, 2001. An introduction with applications to uniformization theorems, partition identities and combinatorial number theory. MR 1850752
  • [Fol] A. Folsom, Modular forms and Eisenstein's continued fractions. Preprint, 2004.
  • [Fri] R. Fricke, Lehrbuch der Algebra, Band 2, Vieweg, Braunschweig, 1926.
  • [Göl] H. Göllnitz, Partitionen mit Differenzenbedingungen, J. Reine Angew. Math. 225 (1967), 154–190 (German). MR 0211973, https://doi.org/10.1515/crll.1967.225.154
  • [Gor] Basil Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741–748. MR 0184001
  • [Har] G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work., Chelsea Publishing Company, New York, 1959. MR 0106147
  • [Hei] E. Heine, Verwandlung von Reihen in Kettenbrüche. J. für die reine und angew. Math. 32, 205-209 (1846).
  • [Hir] Michael D. Hirschhorn, On the expansion of Ramanujan’s continued fraction, Ramanujan J. 2 (1998), no. 4, 521–527. MR 1665326, https://doi.org/10.1023/A:1009789012006
  • [Jac] C. G. J. Jacobi, Allgemeine Theorie der kettenbruchaehnlichen Algorithmen, in welchen jede Zahl aus drei vorhergehenden gebildet wird, J. für die reine und angew. Math. 69, 29-64 (1869) [in Mathematische Werke, VI, pp. 385-426, Chelsea, New York, 1969].
  • [Kl1] F. Klein, Weitere Untersuchungen über das Ikosaeder. Math. Ann. 12, 509-561 (1877) [in Gesammelte Math. Abhandlungen II , 321-384, Springer, 1922].
  • [Kl2] F. Klein, Über die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades. Math. Ann. 14, 111-172 (1878) [in Gesammelte Math. Abhandlungen III, 13-75, Springer, 1922].
  • [Kl3] F. Klein, Über die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14, 428-471 (1879) [in Gesammelte Math. Abhandlungen III, 90-136, Springer, 1922].
  • [Kl4] Felix Klein, Lectures on the icosahedron and the solution of equations of the fifth degree, Second and revised edition, Dover Publications, Inc., New York, N.Y., 1956. Translated into English by George Gavin Morrice. MR 0080930
  • [Kno] Marvin I. Knopp, Modular functions in analytic number theory, Markham Publishing Co., Chicago, Ill., 1970. MR 0265287
  • [Mag] Wilhelm Magnus, Vignette of a cultural episode, Studies in numerical analysis (papers in honour of Cornelius Lanczos on the occasion of his 80th birthday), Academic Press, London, 1974, pp. 7–13. MR 0347521
  • [MM] Henry McKean and Victor Moll, Elliptic curves, Cambridge University Press, Cambridge, 1997. Function theory, geometry, arithmetic. MR 1471703
  • [Mu1] T. Muir, New general formula for the transformation of infinite series into continued fractions. Transactions of the Royal Society of Edinburgh, 27 (1876), 467-471.
  • [Mu2] T. Muir, On Eisenstein's continued fractions, Transactions of the Royal Society of Edinburgh, 28 (1876-1878), 135-144.
  • [New] Morris Newman, Classification of normal subgroups of the modular group, Trans. Amer. Math. Soc. 126 (1967), 267–277. MR 0204375, https://doi.org/10.1090/S0002-9947-1967-0204375-3
  • [Ogg] A. P. Ogg, Rational points of finite order on elliptic curves, Invent. Math. 12 (1971), 105–111. MR 0291084, https://doi.org/10.1007/BF01404654
  • [Per] Oskar Perron, Die Lehre von den Kettenbrüchen, Chelsea Publishing Co., New York, N. Y., 1950 (German). 2d ed. MR 0037384
  • [Ra1] K. G. Ramanathan, On Ramanujan’s continued fraction, Acta Arith. 43 (1984), no. 3, 209–226. MR 738134
  • [Ra2] K. G. Ramanathan, On some theorems stated by Ramanujan, Number theory and related topics (Bombay, 1988) Tata Inst. Fund. Res. Stud. Math., vol. 12, Tata Inst. Fund. Res., Bombay, 1989, pp. 151–160. MR 1441329
  • [Ram1] S. Ramanujan, Collected Papers, Chelsea, New York, 1962.
  • [Ram2] Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904
  • [Ram3] Srinivasa Ramanujan, The lost notebook and other unpublished papers, Springer-Verlag, Berlin; Narosa Publishing House, New Delhi, 1988. With an introduction by George E. Andrews. MR 947735
  • [Ro1] L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc. 25 (1894), 318-343.
  • [Ro2] L. J. Rogers, On a type of modular relation, Proc. London Math. Soc. (2) 19, 387-397 (1921).
  • [S] Th. Schneider, Arithmetische Untersuchungen elliptischer Integrale, Math. Ann. 113, 1-13 (1937).
  • [Sc] Bruno Schoeneberg, Elliptic modular functions: an introduction, Springer-Verlag, New York-Heidelberg, 1974. Translated from the German by J. R. Smart and E. A. Schwandt; Die Grundlehren der mathematischen Wissenschaften, Band 203. MR 0412107
  • [Sch] I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, Berl. Ber., 302-321 (1917) [#28 in Gesammelte Abhandlungen, Band II, 117-136, Springer-Verlag, 1973].
  • [Se1] A. Selberg, Über einige arithmetische Identitäten. Avh. Norske Vidensk.- Akad. Oslo I, 1936, Nr. 8, 23 S. (1936) [#1 in Collected papers, Vol. I. With a foreword by K. Chandrasekharan. Springer-Verlag, Berlin, 1989].
  • [Se2] Atle Selberg, Reflections around the Ramanujan centenary [ MR1001304 (90g:01055)], Ramanujan: essays and surveys, Hist. Math., vol. 22, Amer. Math. Soc., Providence, RI, 2001, pp. 203–213. MR 1862753
    Atle Selberg, Reflections around the Ramanujan centenary, Normat 37 (1989), no. 1, 2–7, 43 (Norwegian, with English summary). MR 1001304
  • [Ser1] Jean-Pierre Serre, Extensions icosaédriques, Seminar on Number Theory, 1979–1980 (French), Univ. Bordeaux I, Talence, 1980, pp. Exp. No. 19, 7 (French). MR 604216
  • [Ser2] Jean-Pierre Serre, L’invariant de Witt de la forme 𝑇𝑟(𝑥²), Comment. Math. Helv. 59 (1984), no. 4, 651–676 (French). MR 780081, https://doi.org/10.1007/BF02566371
  • [Ser3] Jean-Pierre Serre, Cohomological invariants, Witt invariants, and trace forms, Cohomological invariants in Galois cohomology, Univ. Lecture Ser., vol. 28, Amer. Math. Soc., Providence, RI, 2003, pp. 1–100. Notes by Skip Garibaldi. MR 1999384
  • [Shi] Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
  • [Sil] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368
  • [Sta] H. M. Stark, On complex quadratic fields wth class-number two, Math. Comp. 29 (1975), 289–302. Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. MR 0369313, https://doi.org/10.1090/S0025-5718-1975-0369313-X
  • [Syl] J. J. Sylvester, On a remarkable modification of Sturm's theorem. Philosophical Magazine 4, 446-457 (1853) [#61 in Collected Math. Papers, Vol. I, 609-619, Chelsea, New York, 1973].
  • [Tot] Gabor Toth, Finite Möbius groups, minimal immersions of spheres, and moduli, Universitext, Springer-Verlag, New York, 2002. MR 1863996
  • [Wa1] G. N. Watson, Theorems stated by Ramanujan (VII): Theorems on continued fractions, J. London Math. Soc. 4 (1929), 39-48.
  • [Wa2] G. N. Watson, Theorems stated by Ramanujan (IX): Two continued fractions, J. London Math. Soc. 4 (1929), 231-237.
  • [Web] H. Weber, Lehrbuch der Algebra, III. Braunschweig, 1908 [reprinted by Chelsea, New York, 1961].

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11Fxx, 11Gxx

Retrieve articles in all journals with MSC (2000): 11Fxx, 11Gxx


Additional Information

W. Duke
Affiliation: Department of Mathematics, University of California, Box 951555, Los Angeles, California 90095-1555
Email: wdduke@ucla.edu

DOI: https://doi.org/10.1090/S0273-0979-05-01047-5
Received by editor(s): November 4, 2003
Published electronically: January 25, 2005
Additional Notes: Research supported in part by NSF Grant DMS-0355564.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.