Degenerations of $q$-Heun equation

We obtain several degenerations of the $q$-Heun equation by considering the linear $q$-difference equations associated to several $q$-Painlevé equations. We establish definitions of the confluent $q$-Heun equation, the biconfluent $q$-Heun equation and the doubly confluent $q$-Heun equation, and investigate limit procedures to the corresponding differential equations.