Georgian Mathematical Journal 13 (2006), No. 4, 793--805
Copyright Heldermann Verlag 2006
On a Double Series of Chan and Ong
Kenneth S. Williams
Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada
kwilliam@connect.carleton.ca
[Abstract-pdf]
An arithmetic identity is used to prove a relation satisfied by the double series $\sum_{m,n=-\infty}^{\infty}q^{m^{2}+mn+2n^{2}}$. As an application an explicit formula is given for the number of representations of the positive integer $n$ by the form $x_{1}^{2} + x_{1}x_{2} + 2x_{2}^{2} + x_{3}^{2} + x_{3}x_{4} + 2x_{4}^{2} + x_{5}^{2} + x_{5}x_{6}+2x_{6}^{2}+x_{7}^{2}+x_{7}x_{8}+2x_{8}^{2}.$
Keywords: Eisenstein series, theta series, quadratic forms in 4 and 8 variables.
MSC: 11F27, 11E25
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