An effective formula for the Bergman kernel on $\mathbb{ H}_{ \gamma} = \{ |z_1|^\gamma < |z_2| < 1 \}$ is obtained for rational $\gamma = \frac{ m}{ n} >1$. The formula depends on arithmetic properties of $\gamma$, which uncovers new symmetries and clarifies previous results. The formulas are then used to study the Lu Qi-Keng problem. We produce sequences of rationals $\gamma_j \searrow 1$, where each $\mathbb{ H}_{ \gamma_j}$ has a Bergman kernel with zeros (while $\mathbb{ H}_1$ is known to have a zero-free kernel), resolving an open question on this domain class.
Categories: math.CV