A general quantum Laplace transform

In this paper, we introduce a general quantum Laplace transform L β and some of its properties associated with the general quantum difference operator D β f ( t ) = ( f ( β ( t ) ) − f ( t ) ) / ( β ( t ) − t ) , β is a strictly increasing continuous function. In addition, we compute the β -Laplace transform of some fundamental functions. As application we solve some β -difference equations using the β -Laplace transform. Finally, we present the inverse β -Laplace transform L β − 1 .