Hahn introduced the difference operator Dq,ωf(t)=(f(qt+ω)−f(t))/(t(q−1)+ω) in 1949, where 0<q0 are fixed real numbers. This operator extends the classical difference operator Δωf(t)=(f(t+ω)−f(t))/ω as the Jackson q-difference operator Dqf(t)=(f(qt)−f(t))/(t(q−1)).

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

In this paper, we revisit the \(L^p\)-spaces, \(p\geq 1\), associated with a general quantum difference operator and prove some convergence theorems in the quantum setting. Furthermore, two inequalities of Hardy's type are established. Finally, many illustrative examples concerning with \(q\)-difference operator, Hahn difference operator and power quantum difference operator are given.

We present several new sufficient conditions for Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear dynamic equations for functions defined on a time scale with values in a Banach space.