LAUSR

In this article, we study the complex symmetry of composition operators $$C_{\phi }f=f\circ \phi $$ induced on the weighted Bergman spaces $$A^2_{\beta }(\mathbb {D}),$$ by analytic self-maps of the unit disk. One of our main results shows that if $$C_\phi $$ is complex symmetric then $$\phi $$ must fix a point in $$\mathbb {D}$$ . From this, we prove that if $$\phi $$ is neither constant nor an elliptic automorphism of $$\mathbb {D}$$ and $$C_{\phi }$$ is complex symmetric then $$C_{\phi }$$ and $$C_{\phi }^*$$ are cyclic operators. Moreover, by assuming $$\phi $$ is an elliptic automorphism of $$\mathbb {D}$$ which not a rotation and $$\beta \in \mathbb {N},$$ we show that $$C_{\phi }$$ is not complex symmetric whenever $$\phi $$ has order greater than $$2(3+\beta ).$$