Quadratic forms and a product-to-sum formula

Kenneth S. Williams¹

¹ Centre for Research in Algebra and Number Theory School of Mathematics and Statistics Carleton University Ottawa, Ontario, Canada K1S 5B6

Acta Arithmetica, Tome 158 (2013) no. 1, pp. 79-97

Résumé

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 \[ \Bigl[q\prod_{k=1}^{\infty}(1-q^{4k})^6 \Bigr]_n=\sum_{\textstyle{(x_1,x_2)\in \mathbb Z^2\atop x_1^2+4x_2^2=n}}\frac12(x_1^2-4x_2^2). \]

Détail

DOI : 10.4064/aa158-1-5

Keywords: mathbb satisfy sum infty n write prove general product to sum formula which includes known formulae bigl prod infty q q bigr sum textstyle mathbb atop frac bigl prod infty q bigr sum textstyle mathbb atop frac