The Bergman theory of domains ${ |{z_{1} |^{gamma}} < |{z_{2}} | < 1 }$ in $mathbb{C}^2$ is studied for certain values of $gamma$, including all positive integers. For such $gamma$, we obtain a closed form expression for the Bergman kernel, $mathbb{B
}_{gamma}$. With these formulas, we make new observations relating to the Lu Qi-Keng problem and analyze the boundary behavior of $mathbb{B}_{gamma}(z,z)$.
