Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

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

 
 

 

Euler’s “De Partitio Numerorum”


Author: George E. Andrews
Journal: Bull. Amer. Math. Soc. 44 (2007), 561-573
MSC (2000): Primary 11P81, 11P83, 05A17, 05A19
DOI: https://doi.org/10.1090/S0273-0979-07-01180-9
Published electronically: June 18, 2007
MathSciNet review: 2338365
Full-text PDF Free Access

References | Similar Articles | Additional Information

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

  • H. L. Alder, Partition identities—from Euler to the present, Amer. Math. Monthly 76 (1969), 733–746. MR 263766, DOI https://doi.org/10.2307/2317861
  • Krishnaswami Alladi, The method of weighted words and applications to partitions, Number theory (Paris, 1992–1993) London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 1–36. MR 1345170, DOI https://doi.org/10.1017/CBO9780511661990.002
  • George E. Andrews, On $q$-difference equations for certain well-poised basic hypergeometric series, Quart. J. Math. Oxford Ser. (2) 19 (1968), 433–447. MR 237831, DOI https://doi.org/10.1093/qmath/19.1.433
  • George E. Andrews, Partition identities, Advances in Math. 9 (1972), 10–51. MR 306105, DOI https://doi.org/10.1016/0001-8708%2872%2990028-X
  • George E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR 0557013
  • George E. Andrews, The hard-hexagon model and Rogers-Ramanujan type identities, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 9, 5290–5292. MR 629656, DOI https://doi.org/10.1073/pnas.78.9.5290
  • George E. Andrews, L. J. Rogers and the Rogers-Ramanujan identities, Math. Chronicle 11 (1982), no. 1-2, 1–15. MR 677447
  • George E. Andrews, Euler’s pentagonal number theorem, Math. Mag. 56 (1983), no. 5, 279–284. MR 720648, DOI https://doi.org/10.2307/2690367
  • George E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc. 49 (1984), no. 301, iv+44. MR 743546, DOI https://doi.org/10.1090/memo/0301
  • GEA10 G. E. Andrews, Partitions, from The History of Combinatorics, R. Wilson, ed., to appear.
  • George E. Andrews and R. J. Baxter, Lattice gas generalization of the hard hexagon model. III. $q$-trinomial coefficients, J. Statist. Phys. 47 (1987), no. 3-4, 297–330. MR 894396, DOI https://doi.org/10.1007/BF01007513
  • George E. Andrews, R. J. Baxter, and P. J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Statist. Phys. 35 (1984), no. 3-4, 193–266. MR 748075, DOI https://doi.org/10.1007/BF01014383
  • F. C. Auluck, On some new types of partitions associated with generalized Ferrers graphs, Proc. Cambridge Philos. Soc. 47 (1951), 679–686. MR 45147
  • Richard E. Borcherds, Automorphic forms on ${\rm O}_{s+2,2}({\bf R})$ and infinite products, Invent. Math. 120 (1995), no. 1, 161–213. MR 1323986, DOI https://doi.org/10.1007/BF01241126
  • A. Cayley, Note on a Partition-Series, Amer. J. Math. 6 (1883/84), no. 1-4, 63–64. MR 1505342, DOI https://doi.org/10.2307/2369210
  • Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality., Chelsea Publishing Co., New York, 1966. MR 0245499
  • Leonhard Euler, Introduction to analysis of the infinite. Book I, Springer-Verlag, New York, 1988. Translated from the Latin and with an introduction by John D. Blanton. MR 961255
  • Euler17 L. Euler, Evolutis producti infiniti $(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$ etc., Opera Omnia (1) 3, 472–479.
  • Nathan J. Fine, Basic hypergeometric series and applications, Mathematical Surveys and Monographs, vol. 27, American Mathematical Society, Providence, RI, 1988. With a foreword by George E. Andrews. MR 956465
  • Gollnitz H. Göllnitz, Einfache Partitionen, Diplomarbeit W. S., 1960, Göttingen, 65 pp.
  • Basil Gordon, Some continued fractions of the Rogers-Ramanujan type, Duke Math. J. 32 (1965), 741–748. MR 184001
  • G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge, England; Macmillan Company, New York, 1940. MR 0004860
  • G. H. Hardy, Collected papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others). Vol. I, Clarendon Press, Oxford, 1966. Edited by a committee appointed by the London Mathematical Society. MR 0201267
  • G. H. Hardy, Collected papers of G. H. Hardy (Including Joint papers with J. E. Littlewood and others). Vol. I, Clarendon Press, Oxford, 1966. Edited by a committee appointed by the London Mathematical Society. MR 0201267
  • G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
  • C. G. J. Jacobi, Gesammelte Werke. Bände I–VIII, Chelsea Publishing Co., New York, 1969 (German). Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften. Zweite Ausgabe. MR 0260557
  • Percy A. MacMahon, Combinatory analysis, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR 0141605
  • Ken Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293–307. MR 1745012, DOI https://doi.org/10.2307/121118
  • Ken Ono, The web of modularity: arithmetic of the coefficients of modular forms and $q$-series, CBMS Regional Conference Series in Mathematics, vol. 102, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2004. MR 2020489
  • Rogers1 L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc. London Math. Soc., 25 (1894), 318–343. Rogers2 L. J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. London Math. Soc. (2), 16 (1917), 315–336. RogersRam L. J. Rogers and S. Ramanujan, Proof of certain identities in combinatory analysis (with a prefatory note by G. H. Hardy), Proc. Cambridge Phil. Soc., 19 (1919), 211–216. Schur I. Schur, Ein Beitrag zur additiven zahlentheorie und zur Theorie der Kettenbrüche, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl., 1917, pp. 302–321.
  • Jean-Pierre Serre, Divisibilité de certaines fonctions arithmétiques, Enseign. Math. (2) 22 (1976), no. 3-4, 227–260. MR 434996
  • L. J. Slater, Further identities of the Rogers-Ramanujan type, Proc. London Math. Soc. (2) 54 (1952), 147–167. MR 49225, DOI https://doi.org/10.1112/plms/s2-54.2.147
  • Sylvester J. J. Sylvester, Preuve graphique du theorème d’Euler sur la partition des nombres pentagonaux, Comptes Rendus, XCVI (1883), 743–745 (Reprinted: Coll. Math. Papers, Vol. 4, Cambridge University Press, Cambridge, 1912, pp. 93–94).

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11P81, 11P83, 05A17, 05A19

Retrieve articles in all journals with MSC (2000): 11P81, 11P83, 05A17, 05A19


Additional Information

George E. Andrews
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
MR Author ID: 26060
Email: andrews@math.psu.edu

Keywords: Euler, partitions, pentagonal number theorem, signed partitions
Received by editor(s): April 24, 2007
Published electronically: June 18, 2007
Additional Notes: The author was partially supported by National Science Foundation Grant DMS 0200097
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.