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, some integral inequalities based on the general quantum difference operator D β are deduced. Here, D β is defined by D β f ( t ) = ( f ( β ( t ) ) − f ( t ) ) / ( β ( t ) − t ) , where β is a strictly increasing continuous function, defined on an interval I ⊆ R , that has one fixed point s 0 ∈ I . The β-Hölder and β-Minkowski inequalities are proved. Also, the β-Gronwall, β-Bernoulli, and some related inequalities are shown. Finally, the β-Lyapunov inequality is established.

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

In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations $$a_{0}(t)D_{\beta}^{2}y(t)+a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t),\quad t \in I, $$ $a_{0}(t)\neq0$ , in a neighborhood of the unique fixed point $s_{0}$ of the strictly increasing continuous function β, defined on an interval $I\subseteq{\mathbb{R}}$ . These equations are based on the general quantum difference operator $D_{\beta}$ , which is defined by $D_{\beta}{f(t)}= (f(\beta(t))-f(t) )/ (\beta(t)-t )$ , $\beta(t)\neq t$ . We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.

In this paper, we derive the solutions of homogeneous and non-homogeneous nth-order linear general quantum difference equations based on the general quantum difference operator \(D_{\beta }\) which is defined by \(D_{\beta }{f(t)}= (f(\beta (t))-f(t) )/ (\beta (t)-t )\), \(\beta (t)\neq t\), where β is a strictly increasing continuous function defined on an interval \(I\subseteq \mathbb{R}\) that has only one fixed point \(s_{0}\in {I}\). We also give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of these equations. Furthermore, we present the fundamental set of solutions when the coefficients are constants, the β-Wronskian associated with \(D_{\beta }\), and Liouville’s formula for the β-difference equations. Finally, we introduce the undetermined coefficients, the variation of parameters, and the annihilator methods for the non-homogeneous β-difference equations.