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


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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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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