Sei ${\displaystyle {}z\in \mathbb {C} }$, ${\displaystyle {}\vert {z}\vert <1}$. Zu ${\displaystyle {}j\in \mathbb {Z} }$ sei

${\displaystyle I_{j}={\left\{(k,\ell )\in \mathbb {N} ^{2}\mid k-\ell =j\right\}}.}$

Berechne zu jedem ${\displaystyle {}j\in \mathbb {Z} }$ zur summierbaren Familie

${\displaystyle z^{k}z^{\ell },\,(k,\ell )\in \mathbb {N} ^{2},}$

die Teilsummen

${\displaystyle t_{j}=\sum _{(k,\ell )\in I_{j}}z^{k}z^{\ell }}$

${\displaystyle {}\sum _{j\in \mathbb {Z} }t_{j}}$