A general quantum Laplace transform

In this paper, we introduce a general quantum Laplace transform Lβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{L}_{\beta }$\end{document} and some of its properties associated with the general quantum difference operator Dβf(t)=(f(β(t))−f(t))/(β(t)−t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${D}_{\beta }f(t)= ({f(\beta (t))-f(t)} )/ ({ \beta (t)-t} )$\end{document}, 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 Lβ−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{L}_{\beta }^{-1}$\end{document}.


Introduction
The Laplace transform in continuous and discrete cases has an essential role in applied mathematics and in mathematical physics, particularly in solving differential and difference equations, respectively. Recently, versions of Laplace transform in other calculi, such as q-calculus and time scale, were investigated, see [2][3][4][5]. The q-Laplace transform has a similar role in solving q-difference equations, see [1]. The general quantum difference operator D β is defined in [12] by where the function y is defined on an interval I ⊆ R and β is a strictly increasing continuous general function, that is, β(t) ∈ I for t ∈ I. The function y is said to be β-differentiable if it is classic differentiable at the fixed points of the function β.  [12] established the calculus based on D β when β has only one fixed point s 0 ∈ I that satisfies the inequality (ts 0 )(β(t)t) ≤ 0 for all t ∈ I, accordingly lim k→∞ β k (t) = s 0 , β k (t) := β • β • · · · • β k-times (t). Examples of this type are the Jackson q-difference operator with β(t) = qt, 0 < q < 1, s 0 = 0 and the Hahn difference operator with β(t) = qt + ω, 0 < q < 1, ω > 0, s 0 = ω 1-q . They mentioned also another type of β when it has only one fixed point s 0 ∈ I and satisfies the inequality (ts 0 )(β(t)t) ≥ 0 for all t ∈ I; consequently, lim k→∞ β k (t) = ∞, for example, the backward Hahn difference operator with β(t) = qt + ω, q > 1, ω > 0. A study of different types of the function β according to the number of its fixed points, which can be basis for different calculi, was presented in [16]. In [13] some integral inequalities based on D β were introduced. The homogeneous second-order linear β-difference equations and the theory of nth-order linear β-difference equations were studied in [8,9]. In addition, some properties of the quantum exponential functions in a Banach algebra were studied in [10]. Properties of the β-Lebesgue spaces were introduced in [6]. The β-difference operator D β and its calculus has applications in many areas in mathematics and physics such as the quantum variational calculus, the orthogonal polynomials, quantum mechanics, and scale of relativity, see [7,14,15].
In this paper we deduce a general quantum Laplace transform L β associated with D β , where β has only one fixed point s 0 ∈ I with the inequality (ts 0 )(β(t)t) ≤ 0 for all t ∈ I, which will be useful in solving the β-difference equations. We organize this paper as follows: In Sect. 2, we introduce the needed preliminaries from the β-calculus. In Sect. 3, we present the β-regressive functions and define the "β-circle plus" ⊕ β and the "β-circle minus" β , and some associated relations. And then, we introduce the β-Laplace transform and some of its properties. Furthermore, we compute the β-Laplace transform of some fundamental functions. As application, we give two examples to solve some β-difference equations. Finally, we deduce the inverse β-Laplace transform L -1 β .

Preliminaries
In this section, we introduce some needed preliminaries from the β-calculus, where β has only one fixed point s 0 ∈ I such that (ts 0 )(β(t)t) ≤ 0 for all t ∈ I, X is a Banach space.
Theorem 2.1 ([12]) Assume that f : I → X and g : I → R are β-differentiable functions on I. Then: (i) The product fg : I → X is β-differentiable at t ∈ I and , provided that g(t)g(β(t)) = 0.
provided that the series converges at x = a and x = b. f is called β-integrable on I if the series converges at a and b for all a, b ∈ I. Clearly, if f is continuous at s 0 ∈ I, then f is β-integrable on I.
Here, at least one of the functions f and g is a real-valued function.

Theorem 2.7 ([11])
The β-exponential functions e p,β (t) and E p,β (t) are the unique solutions of the β-initial value problems respectively.

Definition 2.9 ([11])
The β-hyperbolic functions are defined by Theorem 2.10 ([11]) Let p : I → C be a continuous function at s 0 . Then the following properties hold: Theorem 2.11 ([11]) Assume that p, q : I → C are continuous functions at s 0 ∈ I. The following properties are true:

Main results
In this section, we present the β-regressive functions and define the "β-circle plus" ⊕ β and the "β-circle minus" β . We introduce the β-Laplace transform and some of its main properties. Furthermore, we compute the β-Laplace transform of the β-exponential and the β-trigonometric functions. As application, we give two examples to solve some βdifference equations. Finally, we deduce the inverse β-Laplace transform L -1 β .

β-Regressive functions
We denote the set of all β-regressive functions p : I → C and continuous at s 0 by R β , and the set of all β-regressive constants z ∈ C by R c β .
From the definition we conclude that , and (R β , ⊕ β ) form an abelian group. Note that at t = s 0 , ⊕ β and β reduce to the classic addition and subtraction operations. Theorem 3.3 Let p, q ∈ R β , t ∈ I. Then the following statements are true:

The β-Laplace transform
In this section, let sup I = ∞, s 0 ∈ I. We assume that z, β z ∈ R c β and hence e β z,β is well defined. Furthermore, we denote by V ([s 0 , ∞), C) the set of β-integrable functions over each compact subinterval of [s 0 , ∞).
provided this limit exists, and we say that the improper β-integral converges in this case. If this limit does not exist, then we say that the improper β-integral diverges.

Definition 3.7 Suppose f ∈ V ([s 0 , ∞), C). Then the Laplace transform of f is defined by
for all z ∈ R c β for which the β-integral (3.3) exists.
Proof Using Theorem 3.3 (i 5 ), and since Therefore, we have . .

Corollary 3.15
Let f ∈ V ([s 0 , ∞), C) be a function of exponential order λ. Then, for any n ∈ N, we have Proof As a consequence of Theorem 3.14 and using induction, we get Hence, the corollary holds for any n ∈ N.
Example 3.16 Using the β-Laplace transform, find the solution of the β-initial value problem where provided that lim t→∞ F(t)e β z,β (t) = 0.