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Partitions: At the Interface of q-Series and Modular Forms

  • George E. Andrews
Part of the Developments in Mathematics book series (DEVM, volume 10)

Abstract

In this paper we explore five topics from the theory of partitions: (1) the Rademacher conjecture, (2) the Herschel-Cayley-Sylvester formulas, (3) the asymptotic expansions of E.M. Wright, (4) the asymptotics of mock theta function coefficients, (5) modular transformations of q-series.

Key words

partitions q-series modular forms 

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References

  1. 1.
    H.L. Alder, “Partition identities—from Euler to the present,” Amer. Math. Monthly 76 (1969), 733–746.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    G.E. Andrews, “On the theorems of Watson and Dragonette for Ramanujan’s mock theta functions,” Amer. J. Math 88 (1966), 454–490.MathSciNetCrossRefGoogle Scholar
  3. 3.
    G.E. Andrews, “Partition identities,” Advances in Math. 9 (1972), 10–51.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    G.E. Andrews, The Theory of Partitions, Addison-Wesley, Reading 1976; reprinted, Cambridge University Press, Cambridge, 1984, 1998.Google Scholar
  5. 5.
    G.E. Andrews, “Mock theta functions,” Proc. Symp. in Pure Math 49 (1989), 283–298.CrossRefGoogle Scholar
  6. 6.
    J. Arkin, “Researches on partitions,” Duke Math. J. 38 (1970), 403–409.MathSciNetCrossRefGoogle Scholar
  7. 7.
    A.O.L. Atkin, “Proof of a conjecture of Ramanujan,” Glasgow Math. J. 8 (1967), 14–32.MathSciNetCrossRefGoogle Scholar
  8. 8.
    P. Bateman and P. Erdös, “Monotonicity of partition functions,” Mathematika 3 (1956), 1–14.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    R.J. Baxter, “A direct proof of Kim’s identities,” J. Phys. A: Math. Gen. 31 (1998), 1105–1108.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    B.C. Berndt, Ramanujan’s notebooks, Part IV, Springer Verlag, New York, 1994.Google Scholar
  11. 11.
    B.C. Berndt, P.B. Bialek, and A.J. Yee, “Formulas of Ramanujan for the power series coefficients of certain quotients of eisenstein series,” Inter. Math. Res. Not. 21 (2002), 1077–1109.MathSciNetCrossRefGoogle Scholar
  12. 12.
    B.C. Berndt and K. Ono, “Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary,” (Andrews Festschrift, D. Foata, and G.-N. Han, eds.), Springer Verlag, Berlin, 2001, pp. 39–110.Google Scholar
  13. 13.
    A. Cayley, “Researches on the partition of numbers,” Phil. Trans. Royal Soc. 146 (1856), 127–140, (Reprinted: Coll. Math. Papers, 2 (1889), 235–249 ).Google Scholar
  14. A. DeMorgan, “On a new form of difference equation,” Cambridge Math. J. 4 (1843), 87–90.Google Scholar
  15. 15.
    L. Dragonette, “Some asymptotic formulae for the mock theta series of Ramanujan,” Trans. Amer. Math. Soc. 72 (1952), 474–500.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    L. Ehrenpreis, “Function theory for Rogers-Ramanujan-like partition identities,” Contemporary Math 143 (1993), 259–320.MathSciNetCrossRefGoogle Scholar
  17. J.W.L. Glaisher, “On the number of partitions of a number of partitions of a number into a given number of parts,” Quart. J. Pure and Appl. Math. 40 (1908), 57–143.Google Scholar
  18. 18.
    K. Glösel, “Über die Zerlegung der ganzen Zahlen,” Monatschefte Math. Phys. 7 (1896), 133–141.CrossRefzbMATHGoogle Scholar
  19. 19.
    H. Gupta, E.E. Gwyther, and J.C.P. Miller. Tables ofPartitions. Royal Soc. Math. Tables, Cambridge University Press, Cambridge, 1958, Vol. 4.Google Scholar
  20. 20.
    G.H. Hardy and S. Ramanujan, “Asymptotic formulae in combinatory analysis,’ Proc. London Math. Soc. 17(2)(1918),75-II5.Google Scholar
  21. J.F.W. Herschel, “On circulating functions and on the integration of a class of equations of finite differences into which they enter as coefficients.”Phil. Trans. Royal Soc. London 108 (1818), 144-I68.Google Scholar
  22. 22.
    A.E. Ingham, “A Tauberian theorem for partitions” Annals of Math. 42 (1941), 1075–1090.Google Scholar
  23. D. Kim, “Asymmetric XXZ chain at the antiferromagnetic transition: Spectra and partition functions,” J. Phys. A: Math. Gen. 30 (1996), 3817–3836.Google Scholar
  24. 24.
    R. McIntosh, “Some asymptotic formulae for q-hypergeometric series,” J. London Math. Soc. 51 (2) (1995), 120–136.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    P.A. MacMahon, CombinarorvAnalvsis, Cambridge University Press, London, 1916, Vol. 2. reprinted Chelsea, New York, 1960.Google Scholar
  26. 26.
    E. Netto, Lehrbuch der Combinatorik 2nd edn., Teubner. Berlin, 1927; reprinted: Chelsea, New York, 1958.Google Scholar
  27. 27.
    J.L. Nicolas and A. Sârközy, “On the asymptotic behavior of general partition functions,” Ramanujan Journal 4 (2000), 29–39.CrossRefzbMATHGoogle Scholar
  28. 28.
    K. Ono, “On the parity of the partition function in arithmetic progressions.” J. für die r und a. Math. 472 (1996), 1–16.zbMATHGoogle Scholar
  29. 29.
    K. Ono, “Distribution of the partition function modulo m, ’ Annals of Math. 151 (2000), 293–307.CrossRefzbMATHGoogle Scholar
  30. 30.
    P. Paoli, Opuscula Analytica, Liburni, 1780, Opusc. Il ( Meditations Arith. ), Section I.Google Scholar
  31. 31.
    H. Rademacher, Lectures on Elementary Number Theory, Blaisdell, New York, 1964.zbMATHGoogle Scholar
  32. 32.
    H. Rademacher, Topics in Analytic Number Theory, Springer, New York, 1973.CrossRefzbMATHGoogle Scholar
  33. 33.
    S. Ramanujan, Collected Papers,Cambridge University Press, London, 1927; reprinted: A.M.S. Chelsea, 2000 with new preface and extensive commentary by B. Berndt.Google Scholar
  34. 34.
    S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.zbMATHGoogle Scholar
  35. L.B. Richmond, “A general asymptotic result for partitions,” Canadian J. Math. 27 (1975) 1083–1091.Google Scholar
  36. 36.
    K.F. Roth and G. Szekeres, “Some asymptotic formulae in the theory of partitions,” Quant. J. Math. Oxford Series 5 (2) (1954), 241–259.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbriiche, S.-B. Preuss. Akad. Wiss., Phys.-Math. KI., 1926, 488–495.Google Scholar
  38. J.J. Sylvester, “On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order. With an excursus on rational fractions and partitions,” Amer. J. Math. 5 (1882). 79–136.Google Scholar
  39. 39.
    G. Szekeres, “An asymptotic formula in the theory of partitions, II,” Quart. J. Math. Oxford Series 2 (2) (1951), 85–108.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    G.N. Watson, “The final problem: An account of the mock theta functions,” J. London Math. Soc. 11 (1936), 55–80.CrossRefGoogle Scholar
  41. 41.
    G.N. Watson, “Ramanujans Vermutung über Zerfälllungsanzahlen,” J. reine und angel. Math. 179 (1938), 97–128.Google Scholar
  42. 42.
    E.M. Wright, “Asymptotic partition formulae, I. plane partitions.” Quart. J. Math., Oxford Series 2 (1931), 177–189.CrossRefGoogle Scholar
  43. 43.
    E.M. Wright, “Asymptotic partition formulae. II. Weighted partitions,’ Proc. London Math. Soc. 36 (2) (1932), 117–141.Google Scholar
  44. 44.
    E.M. Wright, “Asymptotic partitions formulae, III. Partitions into k-th powers,” Acta Math. 63 (1934), 143–191.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • George E. Andrews
    • 1
  1. 1.The Pennsylvania State UniversityUniversity ParkUSA

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