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Quadratic forms and a product-to-sum formula

Volume 158 / 2013

Kenneth S. Williams Acta Arithmetica 158 (2013), 79-97 MSC: Primary 11E25; Secondary 11F20, 11F25. DOI: 10.4064/aa158-1-5


Let $q \in \mathbb C$ satisfy $|q|<1$. If $f(q)=\sum_{n=0}^{\infty} f_n q^n$ we write $[f(q)]_n=f_n$. We prove a general product-to-sum formula which includes known formulae such as $$ \Bigl[q\prod_{k=1}^{\infty}(1-q^{2k})^3(1-q^{6k})^3 \Bigr]_n =\sum_{\textstyle{(x_1,x_2)\in \mathbb Z^2\atop x_1^2+3x_2^2=n}}\frac12(x_1^2-3x_2^2) $$ and \[ \Bigl[q\prod_{k=1}^{\infty}(1-q^{4k})^6 \Bigr]_n=\sum_{\textstyle{(x_1,x_2)\in \mathbb Z^2\atop x_1^2+4x_2^2=n}}\frac12(x_1^2-4x_2^2). \]


  • Kenneth S. WilliamsCentre for Research in Algebra and Number Theory
    School of Mathematics and Statistics
    Carleton University
    Ottawa, Ontario, Canada K1S 5B6

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