A general quantum difference calculus

Abstract

In this paper, we consider a strictly increasing continuous function β, and we present a general quantum difference operator Dβ$D_{\beta}$ which is defined to be Dβf(t)=(f(β(t))−f(t))/(β(t)−t)${D}_{\beta}f(t)= ({f(\beta(t))-f(t)})/ ({\beta(t)-t})$. This operator yields the Hahn difference operator when β(t)=qt+ω$\beta(t)=qt+\omega$, the Jackson q-difference operator when β(t)=qt$\beta (t)=qt$, q∈(0,1)$q\in(0,1)$, ω>0$\omega>0$ are fixed real numbers and the forward difference operator when β(t)=t+ω$\beta(t)=t+\omega$, ω>0$\omega>{0}$. A calculus based on the operator Dβ$D_{\beta}$ and its inverse is established.