The Ramanujan Journal

, Volume 4, Issue 2, pp 137–155 | Cite as

Powers of Euler's Product and Related Identities

  • Shaun Cooper
  • Michael D. Hirschhorn
  • Richard Lewis

Abstract

Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q)r satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q)r satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q)r(qt; qt)s, for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures.

Ramanujan's partition congruences Macdonald identities Winquist's identity Euler's product 

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References

  1. 1.
    J.H. Conway and S.P. Norton, “Monstrous moonshine,” Bull. London Math. Soc. 11 (1979), 308–339.Google Scholar
  2. 2.
    D. Dummit, H. Kisilevsky, and J. McKay, “Multiplicative products of η-functions,” in Finite groups—coming of age (Montreal, Que., 1982), Contemp. Math. 45, Amer. Math. Soc., Providence, R.I. (1985), 89–98.Google Scholar
  3. 3.
    F.J. Dyson, “Missed opportunities,” Bull. Amer. Math. Soc. 78 (1972), 635–652.Google Scholar
  4. 4.
    N.J. Fine, “Basic hypergeometric series and applications,” Mathematical Surveys and Monographs, Vol. 27. Amer. Math. Soc., Providence, R.I., 1988.Google Scholar
  5. 5.
    B. Gordon and K. Hughes, “Multiplicative properties of ´-products. II,” in A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math. 143 (1993), 415–430.Google Scholar
  6. 6.
    B. Gordon and D. Sinor, “Multiplicative properties of ´-products,” in Number theory, Madras 1987 (K. Alladi, Ed.), Springer Lecture Notes in Math., 1395 (1989), 173–200.Google Scholar
  7. 7.
    G.H. Hardy and E.M. Wright, “An Introduction to the Theory of Numbers,” Oxford, 5th ed., 1989.Google Scholar
  8. 8.
    M.D. Hirschhorn, “A generalisation of Winquist's identity and a conjecture of Ramanujan,” J. Indian Math. Soc. 51 (1987), 49–55.Google Scholar
  9. 9.
    F. Klein and R. Fricke, “Vorlesungen ¨uber die Theorie der elliptische Modulfunktionen,” Vol. 2, Teubner, Leipzig, 1892.Google Scholar
  10. 10.
    I.G. Macdonald, “Affine root systems and Dedekind's-function,” Invent. Math. 15 (1972), 91–143.Google Scholar
  11. 11.
    Y. Martin, “Multiplicative ´-quotients,” Trans. Amer. Math. Soc. 348 (1996), 4825–4856.Google Scholar
  12. 12.
    G. Mason, “Elliptic systems and the eta-function,” Notas Soc. Mat. Chile 8 (1989), 37–53.Google Scholar
  13. 13.
    M. Newman, “An identity for the coefficients of certain modular forms,” J. London Math. Soc. 30 (1955), 488–493.Google Scholar
  14. 14.
    M. Newman”Modular forms whose coefficients possess multiplicative properties,” Ann. of Math. 70(2) (1959), 478–489.Google Scholar
  15. 15.
    M. Newman “Modular forms whose coefficients possess multiplicative properties. II.,” Ann. of Math. 75(2) (1962), 242–250.Google Scholar
  16. 16.
    S. Ramanujan, “On certain arithmetical functions,” Trans. Camb. Phil. Soc. 22 (1916), 159–184.Google Scholar
  17. 17.
    S. Ramanujan, “Some properties of p.(n), the number of partitions of n,” Proc. Camb. Philos. Soc. 19 (1919), 207–210.Google Scholar
  18. 18.
    L. Winquist, “An elementary proof of p.(11n + ) ≡ 0 (mod 11),” J. Combin. Theory, Ser. A 6 (1969), 56–59.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Shaun Cooper
    • 1
  • Michael D. Hirschhorn
    • 2
  • Richard Lewis
    • 3
  1. 1.IIMSMassey University, Albany CampusAucklandNew Zealand
  2. 2.School of MathematicsUNSWSydneyAustralia
  3. 3.SMSUniversity of SussexBrightonUK

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