State complexity of star of union and square of union on k regular languages

In this paper, we investigate the state complexities of and , where Li, 1⩽i⩽k, k⩾2 are regular languages. We establish exact bounds for both of these general combined operations and show that they are much lower than the mathematical compositions of the state complexities of their basic individual component operations, but have similar forms with the state complexities of some participating combined operations.