The Ramanujan Journal

, Volume 6, Issue 3, pp 347–367 | Cite as

Results of Hurwitz Type for Five or More Squares

  • P. Barrucand
  • S. Cooper
  • M.D. Hirschhorn

Abstract

Let rk(n) denote the number of representations of n as a sum of k squares. We give many cases (for k = 5, 6, 7) in which the generating function ∑n≥0rk(an + b)qn is a simple infinite product. For instance,
$$\sum\limits_{n \geqslant 0} {r_7 \left( {24n + 23} \right)q^n } = 49728\mathop \prod \limits_{n \geqslant 1} \frac{{(1 - q^{2n} )^{10} (1 - q^{3n} )^3 }}{{(1 - q^n )^6 }}.$$
sums of squares generating function simple infinite product 

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References

  1. 1.
    S. Cooper and M.D. Hirschhorn, “Results of Hurwitz type for three squares,” submitted.Google Scholar
  2. 2.
    M. Hirschhorn, F. Garvan, and J. Borwein, “Cubic analogues of the Jacobian theta function θ(z, q),” Canad. J. Math. 45 (1993), 673-694.Google Scholar
  3. 3.
    M.D. Hirschhorn, “Jacobi's two-square theorem and related results,” The Ramanujan Journal 3 (1999), 153-158.Google Scholar
  4. 4.
    M.D. Hirschhorn and J.A. McGowan, “Algebraic consequences of Jacobi's two-and four-square theorems,” in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (F.G. Garvan and M.E.H. Ismail, eds.), Kluwer Academic Publishers, Dordrecht, 2001, 107-132.Google Scholar
  5. 5.
    A. Hurwitz, “Ueber die Anzahl der Classen quadratischer Formen von negativer Determinante,” Mathematische Werke, Band II, Birkhauser, Basel (1933), 68-71.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • P. Barrucand
    • 1
  • S. Cooper
    • 2
  • M.D. Hirschhorn
    • 3
  1. 1.ParisFrance
  2. 2.IIMSMassey University, Albany CampusAucklandNew Zealand
  3. 3.School of MathematicsUNSWSydneyAustralia

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