Exponential, Trigonometric and Hyperbolic Functions Associated with

Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 12, Number 1, pp. 25–38 (2017) http://campus.mst.edu/adsa

Exponential, Trigonometric and Hyperbolic Functions Associated with a General Quantum Difference Operator Alaa E. Hamza Cairo University, Faculty of Science Mathematics Department Giza, Egypt [email protected] Abdel-Shakoor M. Sarhan and Enas M. Shehata Menoufia University, Faculty of Science Mathematics Department Shibin El-Koom, Egypt [email protected] and [email protected]

Abstract In this paper, we present the β-exponential and β-trigonometric functions based on the general quantum difference operator Dβ defined by Dβ f (t) =

f (β(t)) − f (t) , β(t) 6= t, β(t) − t

which are the solutions of the first and second order β-difference equations, respectively. Here, β is a strictly increasing continuous function defined on an interval I ⊆ R. Furthermore, we establish many properties of these functions. Finally, the β-hyperbolic functions and their properties are introduced.

AMS Subject Classifications: 39A10, 39A13, 39A70, 47B39. Keywords: Quantum difference operator, quantum calculus, Hahn difference operator, Jackson q-difference operator. Received November 11, 2016; Accepted April 27, 2017 Communicated by Martin Bohner

26

1

A. E. Hamza, A. M. Sarhan and E. M. Shehata

Introduction

Quantum calculus is recently subject to an increase number of investigations, due to its applications. It substitutes the usual derivative by a difference operator, which allows one to deal with sets of non-differentiable functions [3, 10]. Quantum calculus has applications in many areas, for instance calculus of variations, economical systems, and several fields of physics such as black holes, quantum mechanics, nuclear and high energy physics [7–9, 16–19]. In [14] we have constructed a general quantum difference operator Dβ , by considering a strictly increasing continuous function β : I → I, where I is an interval of R containing a fixed point s0 of β. The β-difference operator is defined by  f (β(t)) − f (t)   , t 6= s0 ,  β(t) − t Dβ f (t) =    f 0 (s0 ), t = s0 . where f is an arbitrary function defined on I, and is differentiable at t = s0 in the usual sense. As particular cases, we obtain the Hahn difference operator Dq,ω when β(t) = qt + ω, q ∈ (0, 1), ω > 0 are fixed numbers, the Jackson q-difference operator Dq when β(t) = qt, q ∈ (0, 1), the n, q-power quantum difference operator when β(t) = qtn , q ∈ (0, 1), n is a fixed odd positive integer and the forward difference operator ∆a,b when β(t) = at + b, a ≥ 1, b ≥ 0 and a + b > 1. For more details about these operators we refer the reader to [1, 2, 4–6, 11–13, 15]. In a first step towards the development of the general quantum difference calculus, in [14], we considered our function β when it has only one fixed point s0 ∈ I and satisfies the following condition (t − s0 )(β(t) − t) ≤ 0 for all t ∈ I, and gave a rigorous analysis of the calculus based on Dβ and its associated integral operator. Some basic properties of such a calculus were stated and proved. For instance, the chain rule, Leibniz’ formula, the mean value theorem and the fundamental theorem of β-calculus. This paper is organized as follows. In Section 2, the β-exponential functions are defined and some of their properties are introduced. Also, we prove that they are the unique solutions of the first order β-difference equations. In Section 3, the βtrigonometric functions are presented and their properties are established. In Section 4, the β-hyperbolic functions are exhibited and their properties are shown. Throughout the paper X is a Banach space, I is an interval of R containing only one fixed point s0 of β and β k (t) := β ◦ β ◦ · · · ◦ β (t). | {z } k−times

We need the following results from [14] to prove our main results.

27

Exponential General Quantum Difference Operator Theorem 1.1. If f : I → X is continuous at s0 , then the series

X∞ k=0

||(β k (t) −

β k+1 (t))f (β k (t))|| is uniformly convergent on every compact interval J ⊆ I containing s0 . Definition 1.2. For a function f : I → X, we define the β-difference operator of f as    f (β(t)) − f (t) , t 6= s0 ,  β(t) − t Dβ f (t) =    f 0 (s0 ), t = s0 provided that f 0 exists at s0 . In this case, we say that Dβ f (t) is the β-derivative of f at t. We say that f is β-differentiable on I if f 0 (s0 ) exists. Theorem 1.3. Assume that f : I → X and g : I → R are β-differentiable at t ∈ I. Then: (i) The product f g : I → X is β-differentiable at t and Dβ (f g)(t) = (Dβ f (t))g(t) + f (β(t))Dβ g(t) = (Dβ f (t))g(β(t)) + f (t)Dβ g(t). (ii) f /g is β-differentiable at t and
(Dβ f (t))g(t) − f (t)Dβ g(t) Dβ f /g (t) = , g(t)g(β(t)) provided that g(t)g(β(t)) 6= 0. Lemma 1.4. Let f : I → X be β-differentiable and Dβ f (t) = 0 for all t ∈ I, then f (t) = f (s0 ) for all t ∈ I. Theorem 1.5. Assume f : I → X is continuous at s0 . Then the function F defined by F (t) =

∞ X


β k (t) − β k+1 (t) f (β k (t)), t ∈ I,

(1.1)

k=0

is a β-antiderivative of f with F (s0 ) = 0. Conversely, a β-antiderivative F of f vanishing at s0 is given by the formula (1.1). Definition 1.6. Let f : I → X and a, b ∈ I. We define the β-integral of f from a to b by Z b Z b Z a f (t)dβ t = f (t)dβ t − f (t)dβ t, (1.2) a

s0

s0

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A. E. Hamza, A. M. Sarhan and E. M. Shehata

where Z

x

f (t)dβ t = s0

∞ X


β k (x) − β k+1 (x) f (β k (x)), x ∈ I.

(1.3)

k=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, b for all a, b ∈ I. Clearly, if f is continuous at s0 ∈ I, then f is β-integrable on I. Theorem 1.7. Let f : I → X be continuous at s0 . Define the function Z x F (x) = f (t)dβ t, x ∈ I.

(1.4)

s0

Then F is continuous at s0 , Dβ F (x) exists for all x ∈ I and Dβ F (x) = f (x).

2 β-Exponential Functions In this section, we define the β-exponential functions and we study some of their properties. Definition 2.1 (β-Exponential Functions). Assume that p : I → C is a continuous function at s0 . We define the β-exponential functions ep,β (t) and Ep,β (t) by h ep,β (t) = Q ∞ k=0

and

1 1−

p(β k (t))(β k (t)

−

i

(2.1)

β k+1 (t))

∞ h i Y Ep,β (t) = 1 + p(β k (t))(β k (t) − β k+1 (t)) ,

(2.2)

k=0

It is worth mentioning that both products in (2.1) and (2.2) are convergent since ∞ X

|p(β k (t))(β k (t) − β k+1 (t))|

k=0

is uniformly convergent by Theorem 1.1. For the case when p is a constant function p(t) = z, z ∈ C and β(t) = qt + ω, ω > 0 and q ∈ (0, 1), we obtain the Hahn exponential functions, see [4]. From (2.1), (2.2) we have ep,β (t) =

1 E−p,β (t)

.

(2.3)

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Exponential General Quantum Difference Operator

Theorem 2.2. The β-exponential functions ep,β (t) and Ep,β (t) are the unique solutions of the first order β-difference equations Dβ y(t) = p(t)y(t),

y(s0 ) = 1,

(2.4)

and Dβ y(t) = p(t)y(β(t)),

y(s0 ) = 1,

(2.5)

respectively. Proof. It is obvious that ep,β (s0 ) = Ep,β (s0 ) = 1. We have ep,β (β(t)) − ep,β (t) β(t) − t h 1 1 Q∞ = k+1 (t))(β k+1 (t) − β k+2 (t))) β(t) − t k=0 (1 − p(β i 1 − Q∞ k k k+1 (t))) k=0 (1 − p(β (t))(β (t) − β p(t) = Q∞ = p(t)ep,β (t). k k k+1 (t))) k=o (1 − p(β (t))(β (t) − β

Dβ ep,β (t) =

Similarly, we see that Ep,β (t) is a solution of (2.5). Finally, to prove the uniqueness of the solution ep,β (t), let x(t) be another solution of (2.4). We have  Dβ

x(t) ep,β (t)

 =

ep,β (t)Dβ x(t) − x(t)Dβ ep,β (t) = 0, t ∈ I. ep,β (t)ep,β (β(t))

x(t) x(t) x(s0 ) is a constant function and = = 1, i.e., x(t) = ep,β (t) ep,β (t) ep,β (s0 ) ep,β (t) for all t ∈ I. Similarly Ep,β (t) is the unique solution of (2.5). By Lemma 1.4,

Consider the non-homogeneous first order linear β-difference equation Dβ y(t) = p(t)y(t) + f (t), y(s0 ) = y0 ∈ X.

(2.6)

Theorem 2.3. Let f : I → X be continuous function at s0 . then Z t   y(t) = ep,β (t) y0 + f (τ )E−p,β (β(τ ))dβ τ s0

is a solution of equation (2.6).

(2.7)

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A. E. Hamza, A. M. Sarhan and E. M. Shehata

Proof. We have Z

t

f (τ )E−p,β (β(τ ))dβ τ

Dβ y(t) = Dβ ep,β (t)y0 + Dβ ep,β (t) s0

+ ep,β (β(t))f (t)E−p,β (β(t)) Z t = p(t)ep,β (t)y0 + p(t)ep,β (t) f (τ )E−p,β (β(τ ))dβ τ + f (t) s0

= p(t)y(t) + f (t). Also, y(s0 ) = y0 . In the following two theorems we introduce some important properties of the βexponential function. Theorem 2.4. Let p : I → C be a continuous function at s0 . Then the following properties hold: (i) ep,β (β(t)) = [1 + (β(t) − t)p(t)]ep,β (t), t ∈ I, (ii) Dβ (

(iii)

−p(t) 1 )= , ep,β (t) ep,β (β(t))

1 is the unique solution of the first order β-difference equation ep,β (t) Dβ y(t) =

Proof.

−p(t)ep,β (t) y(t), ep,β (β(t))

y(s0 ) = 1.

(i) From the definition of Dβ , we have ep,β (β(t)) = ep,β (t) + (β(t) − t)Dβ ep,β (t) = ep,β (t)[1 + (β(t) − t)p(t)].

(ii) By Theorem 1.3 (ii), we get Dβ (

−Dβ ep,β (t) 1 −p(t) )= = . ep,β (t) ep,β (t)ep,β (β(t)) ep,β (β(t))

(iii) We can see that 1 = 1. ep,β (s0 )

(2.8)

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Exponential General Quantum Difference Operator

1 is a solution of (2.8). To show that the solution is ep,β (t) unique, suppose that x(t) is another solution of (2.8), then

By part (ii), we get

Dβ (x(t)ep,β (t)) = x(t)Dβ ep,β (t) + Dβ x(t)ep,β (β(t)) p(t)ep,β (t) = x(t)p(t)ep,β (t) − x(t)ep,β (β(t)) = 0. ep,β (β(t)) Hence, x(t)ep,β (t) = x(s0 )ep,β (s0 ) = 1. Therefore, x(t) =

1 . ep,β (t)

Theorem 2.5. Suppose p, q : I → C are continuous functions at s0 . Then the following properties hold: 1 (i) = e −p(t) ,β (t), 1−p(t)(t−β(t)) ep,β (t) (ii) ep,β (t)eq,β (t) = e[p(t)+(β(t)−t)p(t)q(t)+q(t)],β (t), ep,β (t) = e p(t)−q(t) ,β (t). 1−q(t)(t−β(t)) eq,β (t)

(iii) Proof.

e

(i) Clearly, e −p(t) ,β 1−p(t)(t−β(t))

−p(t) ,β 1−p(t)(t−β(t))

(t) is a solution of equation (2.8), then

1 = ep,β (t)

(t).

(ii) We have
Dβ ep,β (t)eq,β (t) = Dβ ep,β (t)eq,β (t) + ep,β (β(t))Dβ eq,β (t) = p(t)ep,β (t)eq,β (t) + q(t)ep,β (β(t))eq,β (t) = p(t)ep,β (t)eq,β (t) + q(t)eq,β (t)[1 + (β(t) − t)p(t)]ep,β (t) = [p(t) + (β(t) − t)p(t)q(t) + q(t)]ep,β (t)eq,β (t). (iii) This is a consequence of (i) and (ii). The proof is complete. 2 1 1 Example 2.6. Let p(t) = and β(t) = t + , for t ∈ [1, 2]. The unique fixed point t 2 2 of the function β is s0 = 1. One can check that 1 k=0 [1 −

e 2 , 1 t+ 1 (t) = Q∞ t 2

2

. t−1 ] t−1+2k

Clearly, e 2 , 1 t+ 1 (1) = 1. So, e 2 , 1 t+ 1 (t) is the unique solution of the equation t 2

2

t 2

2

2 Dβ y(t) = y(t), y(1) = 1. t

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A. E. Hamza, A. M. Sarhan and E. M. Shehata

1 Example 2.7. Let p(t) = t(1 + i) and β(t) = t, for t ∈ [0, 2]. The unique fixed point 2 of the function β is s0 = 0. Clearly, et(1+i), 1 t (0) = 1. One can check that 2

et(1+i), 1 t (t) = 2

1 Q∞

k=0

h

t2 (1 + i) i 1 − 2k+1 2

,

is the unique solution of the equation Dβ y(t) = t(1 + i)y(t), y(0) = 1.

3 β-Trigonometric Functions In this Section we define the β-trigonometric functions and study some of their properties. Definition 3.1 (β-Trigonometric Functions). We define the β-trigonometric functions by eip,β (t) + e−ip,β (t) , (3.1) cosp,β (t) = 2 eip,β (t) − e−ip,β (t) sinp,β (t) = , (3.2) 2i Eip,β (t) + E−ip,β (t) Cosp,β (t) = , (3.3) 2 and Eip,β (t) − E−ip,β (t) Sinp,β (t) = . (3.4) 2i Simple calculations show that the β-trigonometric functions satisfy the relations in the following theorem. Theorem 3.2. For all t ∈ I. The following relations are true: (1) Dβ sinp,β (t) = p(t)cosp,β (t), (2) Dβ cosp,β (t) = −p(t)sinp,β (t), (3) cosp,β (t) + isinp,β (t) = eip,β (t), (4) cos2p,β (t) + sin2p,β (t) = eip,β (t)e−ip,β (t) (at t = s0 , cos2p,β (t) + sin2p,β (t) = 1), (5) Dβ Sinp,β (t) = p(t)Cosp,β (β(t)), (6) Dβ Cosp,β (t) = −p(t)Sinp,β (β(t)), (7) Sin2p,β (t) + Cos2p,β (t) = Eip,β (t)E−ip,β (t),

Exponential General Quantum Difference Operator

33

(8) Cosp,β (t) + iSinp,β (t) = Eip,β (t), (9) sinp,β (t)Sinp,β (t) + cosp,β (t)Cosp,β (t) = 1, (10) sinp,β (t)Cosp,β (t) − cosp,β (t)Sinp,β (t) = 0. In the following theorem it can be easily seen that the β-trigonometric functions are solutions of the second order β-difference equations. Theorem 3.3. Let p : I → C be a continuous function at s0 . Then cosp,β (t), sinp,β (t), Cosp,β (t) and Sinp,β (t) are solutions of the following second order β-difference equations, respectively. i) Dβ2 x(t) = −p2 (t)x(t) − Dβ p(t) sinp,β (β(t)), x(s0 ) = 1, Dβ x(s0 ) = 0. ii) Dβ2 x(t) = −p2 (t)x(t) + Dβ p(t) cosp,β (β(t)), x(s0 ) = 0, Dβ x(s0 ) = p(s0 ). iii)

Dβ2

β 2 (t) − β(t) x(t) = −p (β(t)) x(β 2 (t)) − Dβ p(t) Sinp,β (β(t)), β(t) − t 2

x(s0 ) = 1, Dβ x(s0 ) = 0. iv) Dβ2 x(t) = −p2 (β(t))

β 2 (t) − β(t) x(β 2 (t)) + Dβ p(t) Cosp,β (β(t)), β(t) − t

x(s0 ) = 0, Dβ x(s0 ) = p(s0 ). Corollary 3.4. Let z ∈ C. Then cosz,β (t), sinz,β (t), Cosz,β (t) and Sinz,β (t) are solutions of the following second order β-difference equations, respectively. i) Dβ2 x(t) = −z 2 x(t),

x(s0 ) = 1, Dβ x(s0 ) = 0.

ii) Dβ2 x(t) = −z 2 x(t),

x(s0 ) = 0, Dβ x(s0 ) = z.

iii)

Dβ2

x(t) = −z

iv) Dβ2 x(t) = −z 2

2

(t) − β(t) x(β 2 (t)), β(t) − t

x(s0 ) = 1, Dβ x(s0 ) = 0.

β 2 (t) − β(t) x(β 2 (t)), β(t) − t

x(s0 ) = 0, Dβ x(s0 ) = z.

2β

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A. E. Hamza, A. M. Sarhan and E. M. Shehata

Example 3.5. Let β(t) = qt + ω, q ∈ (0, 1), ω > 0, and let p(t) = z, z ∈ C be a ω β 2 (t) − β(t) constant. Then, s0 = and = q. Consequently, cosz,β (t), sinz,β (t), 1−q β(t) − t Cosz,β (t) and Sinz,β (t) are solutions of the following second order q, ω-difference equations, respectively. 2 i) Dq,ω x(t) = −z 2 x(t),

x(s0 ) = 1, Dq,ω x(s0 ) = 0.

2 x(t) = −z 2 x(t), ii) Dq,ω

x(s0 ) = 0, Dq,ω x(s0 ) = z.

2 x(t) = −z 2 q x(q 2 t + ω(1 + q)), iii) Dq,ω

x(s0 ) = 1, Dq,ω x(s0 ) = 0.

2 iv) Dq,ω x(t) = −z 2 q x(q 2 t + ω(1 + q)),

x(s0 ) = 0, Dq,ω x(s0 ) = z.

Example 3.6. Let β(t) = qt, q ∈ (0, 1), and p(t) = z, z ∈ C be a constant. Then, β 2 (t) − β(t) s0 = 0 and = q. Consequently, cosz,β (t), sinz,β (t), Cosz,β (t) and Sinz,β (t) β(t) − t are solutions of the following second order q, ω-difference equations, respectively. i) Dq2 x(t) = −z 2 x(t),

x(s0 ) = 1, Dq x(0) = 0.

ii) Dq2 x(t) = −z 2 x(t),

x(s0 ) = 0, Dq x(0) = z.

iii) Dq2 x(t) = −z 2 q x(q 2 t), x(s0 ) = 1, Dq x(0) = 0. iv) Dq2 x(t) = −z 2 q x(q 2 t),

x(s0 ) = 0, Dq x(0) = z.

4 β-Hyperbolic Functions In this Section we define the β-hyperbolic functions and study some of their properties. Definition 4.1 (β-Hyperbolic Functions). We define the β-hyperbolic functions by coshp,β (t) =

ep,β (t) + e−p,β (t) , 2

ep,β (t) − e−p,β (t) , 2 Ep,β (t) + E−p,β (t) , Coshp,β (t) = 2 sinhp,β (t) =

and Sinhp,β (t) =

Ep,β (t) − E−p,β (t) . 2

(4.1) (4.2) (4.3)

(4.4)

The following theorem introduces some properties of the β-hyperbolic functions. Its proof is straightforward.

Exponential General Quantum Difference Operator

35

Theorem 4.2. The β-hyperbolic functions satisfy the following properties: (1) Dβ coshp,β (t) = p(t)sinhp,β (t), (2) Dβ sinhp,β (t) = p(t)coshp,β (t), (3) cosh2p,β (t) − sinh2p,β (t) = ep,β (t)e−p,β (t) (at t = s0 , cosh2p,β (t) − sinh2p,β (t) = 1), (4) coshp,β (t) + sinhp,β (t) = ep,β (t), (5) coshp,β (t) − sinhp,β (t) = e−p,β (t), (6) Dβ Coshp,β (t) = p(t)Sinhp,β (β(t)), (7) Dβ Sinhp,β (t) = p(t)Coshp,β (β(t)), (8) Cosh2p,β (t) − Sinh2p,β (t) = Ep,β (t)E−p,β (t), (9) Coshp,β (t) + Sinhp,β (t) = Ep,β (t), (10) Coshp,β (t) − Sinhp,β (t) = E−p,β (t). In the following Theorem, simple calculations show that the β-hyperbolic functions are solutions of second order β-difference equations. Theorem 4.3. Let p : I → C be a continuous function at s0 . Then coshp,β (t), sinhp,β (t), Coshp,β (t) and Sinhp,β (t) are solutions of the following second order β-difference equations, respectively. i) Dβ2 x(t) = p2 (t)x(t) + Dβ p(t) sinhp,β (β(t)), x(s0 ) = 1, Dβ x(s0 ) = 0. ii) Dβ2 x(t) = p2 (t)x(t) + Dβ p(t) coshp,β (β(t)), x(s0 ) = 0, Dβ x(s0 ) = p(s0 ). iii)

Dβ2

β 2 (t) − β(t) x(t) = p (β(t)) x(β 2 (t)) + Dβ p(t) Sinhp,β (β(t)), β(t) − t 2

x(s0 ) = 1, Dβ x(s0 ) = 0. iv) Dβ2 x(t) = p2 (β(t))

β 2 (t) − β(t) x(β 2 (t)) + Dβ p(t) Coshp,β (β(t)), β(t) − t x(s0 ) = 0, Dβ x(s0 ) = p(s0 ).

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A. E. Hamza, A. M. Sarhan and E. M. Shehata

Corollary 4.4. Let z ∈ C. Then coshz,β (t), sinhz,β (t), Coshz,β (t) and Sinhz,β (t) are solutions of the following second order β-difference equations, respectively. i) Dβ2 x(t) = z 2 x(t),

x(s0 ) = 1, Dβ x(s0 ) = 0.

ii) Dβ2 x(t) = z 2 x(t),

x(s0 ) = 0, Dβ x(s0 ) = z.

β 2 (t) − β(t) x(β 2 (t)), β(t) − t

x(s0 ) = 1, Dβ x(s0 ) = 0.

β 2 (t) − β(t) x(β 2 (t)), β(t) − t

x(s0 ) = 0, Dβ x(s0 ) = z.

iii) Dβ2 x(t) = z 2 iv) Dβ2 x(t) = z 2

Example 4.5. Let β(t) = qt + ω, q ∈ (0, 1), ω > 0, and let p(t) = z, z ∈ C be β 2 (t) − β(t) a constant. Then, = q. Consequently, coshz,β (t), sinhz,β (t), Coshz,β (t) β(t) − t and Sinhz,β (t) are solutions of the following second order q, ω-difference equations, respectively. 2 x(t) = z 2 x(t), i) Dq,ω

x(s0 ) = 1, Dq,ω x(s0 ) = 0.

2 ii) Dq,ω x(t) = z 2 x(t),

x(s0 ) = 0, Dq,ω x(s0 ) = z.

2 iii) Dq,ω x(t) = z 2 q x(q 2 t + ω(1 + q)),

x(s0 ) = 1, Dq,ω x(s0 ) = 0.

2 iv) Dq,ω x(t) = z 2 q x(q 2 t + ω(1 + q)),

x(s0 ) = 0, Dq,ω x(s0 ) = z.

Example 4.6. Let β(t) = qt, q ∈ (0, 1) and p(t) = z, z ∈ C be a constant. Then, β 2 (t) − β(t) = q. Consequently, coshz,β (t), sinhz,β (t), Coshz,β (t) and Sinhz,β (t) are β(t) − t solutions of the following second order q-difference equations, respectively. i) Dq2 x(t) = z 2 x(t),

x(s0 ) = 1, Dq x(0) = 0.

ii) Dq2 x(t) = z 2 x(t),

x(s0 ) = 0, Dq x(0) = z.

iii) Dq2 x(t) = z 2 q x(q 2 t),

x(s0 ) = 1, Dq x(0) = 0.

iv) Dq2 x(t) = z 2 q x(q 2 t),

x(s0 ) = 0, Dq x(0) = z.

Conclusion In this paper, we defined the β-exponential functions and proved that they are a unique solutions for the first order β-difference equations. Also, the β-trigonometric functions and their properties were introduced and that they satisfy the second order β-difference equations. Finally, the β-hyperbolic functions and their properties were introduced.

Exponential General Quantum Difference Operator

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Acknowledgments The authors sincerely thank the referees for their valuable suggestions and comments.

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