Georgian Mathematical Journal 13 (2006), No. 4, 675--686
Copyright Heldermann Verlag 2006
Sums of Squares and Sums of Triangular Numbers
Shaun Cooper
Inst. Information Math. Sciences, Massey University, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand
s.cooper@massey.ac.nz
Michael Hirschhorn
School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
m.hirschhorn@unsw.edu.au
[Abstract-pdf]
Motivated by two results of Ramanujan, we give a family of 15 results and 4 related ones. Several have interesting interpretations in terms of the number of representations of an integer by a quadratic form $\lambda_1 x_1^2 + \cdots + \lambda_m x_m^2$, where $\lambda_1 + \cdots +\lambda_m=2$, $4$ or $8$. We also give a new and simple combinatorial proof of the modular equation of order seven.
Keywords: Sum of squares, sum of triangular numbers, theta function, modular equation.
MSC: 11E25; 05A19, 11D85, 33D15
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