A general quantum Laplace transform

Abstract

Abstract In this paper, we introduce a general quantum Laplace transform L β $\mathcal{L}_{\beta }$ and some of its properties associated with the general quantum difference operator D β f ( t ) = ( f ( β ( t ) ) − f ( t ) ) / ( β ( t ) − t ) ${D}_{\beta }f(t)= ({f(\beta (t))-f(t)} )/ ({ \beta (t)-t} )$ , 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 $\mathcal{L}_{\beta }^{-1}$ .

Keywords

• General quantum difference operator
• β-Laplace transform
• Inverse β-Laplace transform
• β-difference equations