Quadratic forms and a product-to-sum formula
Tom 158 / 2013
Acta Arithmetica 158 (2013), 79-97
MSC: Primary 11E25; Secondary 11F20, 11F25.
DOI: 10.4064/aa158-1-5
Streszczenie
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 \[ \Bigl[q\prod_{k=1}^{\infty}(1-q^{4k})^6 \Bigr]_n=\sum_{\textstyle{(x_1,x_2)\in \mathbb Z^2\atop x_1^2+4x_2^2=n}}\frac12(x_1^2-4x_2^2). \]
