International Journal of Mathematical Analysis Vol. 9, 2015, no. 55, 2733 - 2747 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.510243

Leibniz’s Rule and Fubini’s Theorem Associated with Power Quantum Difference Operators

Alaa E. Hamza

Department of Mathematics Faculty of Science Cairo University, Giza, Egypt

M. H. Al-Ashwal

Department of Mathematics Faculty of Science Cairo University, Giza, Egypt

Copyright c 2015 Alaa E. Hamza and M. H. Al-Ashwal. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Jackson in 1908 introduced the well–known and the most used quan- tum difference operator Dq f(t) = (f(qt) − f(t))/(qt − t) for a fixed 0 < q < 1. Aldwoah in 2009 introduced the power quantum n, q– n n difference operator Dn,qf(t) = (f(qt ) − f(t))/(qt − t), where n is an odd natural number and 0 < q < 1 is fixed. Dn,q yields Jackson q– difference operator, when n = 1. In this paper, we establish Leibniz’s rule and Fubini’s theorem associ- ated with this power quantum difference operator.

Mathematics Subject Classification: 39A13, 39A70

Keywords: n, q-power difference operator, n, q–Integral, n, q–Leibniz,s Rule, n, q–Fubini’s Theorem. 2734 Alaa E. Hamza and M. H. Al-Ashwal

1 Introduction

The power quantum difference operator is defined by f(qtn) − f(t)  n t ∈ R\{−θ, 0, θ}, Dn,q f(t) = qt − t f 0 (t) t ∈ {−θ, 0, θ}, where n is a fixed odd positive integer, 0 < q < 1 is fixed number and θ = ∞ 1 for n = 1 and θ = q 1−n for n > 1, which was introduced in [3]. See also [1, 2]. If Dn,q f(t) exists, then f(t) is called n, q–differentiable for all t ∈ R. Here f is supposed to be defined on a set A ⊆ R for which qtn ∈ A whenever t ∈ A. This operator unifies and generalizes two difference operators. The first is the well-known and the most used Jackson q–difference operator which was defined by f(qt) − f(t) D f(t) = , t 6= 0, q t(q − 1) where 0 < q < 1 is fixed. Here f is supposed to be defined on a q-geometric set A, i.e., A is a subset of R (or C) for which qt ∈ A whenever t ∈ A. The derivative at zero is normally defined to be f 0(0), provided that f 0(0) exists [4, 5, 6, 7]. The second operator is the n–power difference operator f(tn) − f(t)  n t ∈ R \ {−1, 0, 1}, Dn f(t) = t − t f 0(t) t ∈ {−1, 0, 1}, where n is a fixed odd positive integer [3]. In [2], Aldwoah et al. gave a rigorous analysis of the calculus associated with Dn,q. They stated and proved some basic properties of such a calculus. For instance, they defined the inverse of Dn,q which contains the right inverse of Dq and the right inverse of Dn. Then, they proved a fundamental lemma of the power quantum variational calculus. This paper is devoted to establishing Leibniz’s rule and Fubini’s theorem associated with the power quantum difference operator. We organize this pa- per as follows. Section 2 gives an introduction to power quantum difference calculus. In Section 3, we prove Leibniz’s rule which is concerning with differ- entiating under the integral sign. Some related results are obtained. Also, we prove Fubini’s theorem in the power quantum difference operator setting, that is, we prove that the iterated integrals are equal.

2 Preliminaries

We assume that I is the interval (−θ, θ) and X is a Banach space, endowed 1 with a norm k · k, where θ = ∞ for n = 1 and θ = q 1−n for n > 1. An essential n, q–Leibniz’s rule and n, q–Fubini’s theorem 2735

y O y=qtn y=t

θ

b h4(b) o a / t −θ p hp5(a) θ

−θ



Figure 1: The iteration of h(t) = qtn, t ∈ I for fixed n > 1 and 0 < q < 1. function which plays an important role in this calculus is h(t) := qtn. The set of fixed points of h(t) is {0} when n = 1 and is {−θ, 0, θ} for n > 1. We study this function on the interval I. One can see that the k–th order iteration of h(t) is given by k hk(t) = q[k]n tn , t ∈ I, where, for α ∈ C,[k]α is defined by

(Pk−1 i i=0 α k ∈ N, [k]α = 0 k = 0.

k ∞ The sequence {h (t)}k=0, see Figure 1, is uniformly convergent to 0 on I.

Theorem 2.1 ([2]). Let f, g : I → R be n, q–differentiable at t ∈ I and c1, c2 ∈ R. Then,
(i) Dn,q c1f + c2 g (t) = c1Dn,q f(t) + c2Dn,q g(t).

n (ii) Dn,q fg (t) = Dn,q f(t) g(t) + f(qt )Dn,q g(t). f  D f(t) g(t) − f(t)D g(t) (iii) D (t) = n,q n,q provided that g(t) g(qtn) 6= n,q g g(t) g(qtn) 0.

We notice that (ii) and (iii) are also true for f : I → X. Also, (i) is true if f, g : I → X. We need the following prelimiary results in our study 2736 Alaa E. Hamza and M. H. Al-Ashwal

Lemma 2.2. Let h(t) be the function defined above. Then, the sequence k of iteration functions {h (t)}k∈N0 converges uniformly to 0 on every interval J = [a, b] ⊆ I containing 0.

k+1 k Proof. For t ∈ [0, b] we have h (t) ≤ h (t) for all k ∈ N0. So, the sequence k k {h (t)}k∈N0 is decreasing to 0. By Dini’s theorem {h (t)}k∈N0 is uniformly convergent to 0 on the interval [0, b]. Similarly, we can prove its uniform k convergence on [a, 0]. Consequently, the sequence {h (t)}k∈N0 is uniformly convergent on the interval J = [a, b]. Corollary 2.3. The series

∞ X k k k [k]n n n n (n−1) q t (q t − 1) k=0 is uniformly convergent to |t| on every interval J = [a, b] ⊆ I containing 0.

k k k Pm [k]n n n n (n−1) Proof. We apply Dini’s theorem to Sm(t) = k=0 −q t (q t − 1) on both [0, b] and [a, 0], then we get the desired result. The proof of the following lemma is a result of continuity and will be omitted.

Lemma 2.4. Let f : I → X be continuous at 0. Then, the sequence k {f(h (t))}k∈N0 converges uniformly to f(0) on every compact interval J ⊆ I containing 0. We need the following theorem to guarantee the convergence of the series in the definition 2.7 of n, q–integral.

Theorem 2.5. If the function f : I → X is continuous at 0, then the series ∞ X k k k k [k]n n n n (n−1) [k]n n q t (q t − 1)f(q t ) k=0 is uniformly convergent on every compact interval J ⊆ I containing 0. Proof. Let J ⊆ I be a compact interval containing 0. By Lemma 2.4, there exists k0 ∈ N such that

k [k]n n f(q t ) − f(0) < 1 ∀t ∈ J, k ≥ k0.

k [k]n n Then f(q t ) < 1 + kf(0)k for k ≥ k0 and t ∈ J which in turn implies that

k k k k k k k [k]n n n n (n−1) [k]n n [k]n n n n (n−1) q t (q t − 1) f(q t ) < q t (q t −1) (1 + kf(0)k) n, q–Leibniz’s rule and n, q–Fubini’s theorem 2737

∀t ∈ J, k ≥ k0. Consider the sequences

m X k k k k [k]n n n n (n−1) [k]n n Gm(t) = q t (q t − 1)f(q t ) k=0 and m X k k k [k]n n n n (n−1) Cm(t) = q t (q t − 1) (1 + kf(0)k). k=0

By Corollary 2.3, Cm(t) is uniformly convergent to |t|(1 + kf(0)k) on J. By Cauchy criterion, given  > 0, there exists m0 ∈ N such that

0 kCm(t) − Cm0 (t)k <  ∀t ∈ J, m ≥ m ≥ m0. This implies that

0 kGm(t) − Gm0 (t)k ≤ kCm(t) − Cm0 (t)k <  ∀t ∈ J, m ≥ m ≥ max{m0, k0}.

∞ X k k k k [k]n n n n (n−1) [k]n n Therefore, q t (q t − 1)f(q t ) is uniformly convergent on k=0 J.

Theorem 2.6 (power quantum chain rule). Assume g : I → R is continuous and n, q–differentiable and f : R → R is continuously differentiable. Then there exists a constant c between qtn and t with

Dn,q(f ◦ g)(t) = f (g(c)) Dn,q g(t). (1) Relation (1) is true at t = 0, by the classical chain rule.

Definition 2.7 ([2]). Assume that f : I → X is a function and a, b ∈ I. The n, q–integral of f from a to b is defined by Z b Z b Z a f(t) dn,qt = f(t) dn,qt − f(t) dn,qt, a 0 0 where

x ∞ Z k k k k X [k]n n n n (n−1) [k]n n f(t) dn,qt = − q x (q x − 1)f(q x ), x ∈ I, (2) 0 k=0 provided that the series converges at x = a and x = b. In this case, we say that f is n, q–integrable on [a, b]. If f is continuous at 0, then the series (2) is convergent. The following results are proved in [2]. 2738 Alaa E. Hamza and M. H. Al-Ashwal

Lemma 2.8. Let f, g : I → R be n, q–integrable, k1, k2 ∈ R and a, b, c ∈ I. Then, Z a (i) f(t) dn,qt = 0. a Z b Z a (ii) f(t) dn,qt = − f(t) dn,qt. a b Z b Z b Z b
(iii) k1f + k2 g (t) dn,qt = k1 f(t) dn,qt + k2 g(t) dn,qt. a a a Z b Z c Z b (iv) f(t) dn,qt = f(t) dn,qt + f(t) dn,qt for a ≤ c ≤ b. a a c Theorem 2.9. Assume that f : I → R is continuous at 0. Define Z x F (x) := f(t) dn,qt, x ∈ I. 0

Then, F is continuous at 0. Furthermore, Dn,q F (x) exists for every x ∈ I and

Dn,q F (x) = f(x).

Conversely,

Z b Dn,q f(t) dn,qt = f(b) − f(a) for all a, b ∈ I. a

Theorem 2.10. Let f : I → R be continuous function at 0, then for t ∈ I Z qtn n f(s) dn,qs = (qt − t)f(t). t 3 Power Quantum Differentiation Under The Integral Sign

In this section we study the continuity and the power quantum differentiation of the integral Z ψ(t) f(t, s) dn,qs. φ(t) We establish Leibniz’s rule. Finally, we prove that the iterated integrals are equal (this theorem is known by Fubini’s Theorem). Let f : I × I → R. We begin by the following definitions. n, q–Leibniz’s rule and n, q–Fubini’s theorem 2739

Definition 3.1. (i) We say that f(t, s) is continuous at t = t0 uniformly with respect to s ∈ A ⊆ I if

lim f(t, s) = f(t0, s) uniformly with respect to s ∈ A. t→t0

(ii) The n, q–partial derivative of f with respect to t is defined by

f(qtn, s) − f(t, s)   n t ∈ I\{0}, D f(t, s) = qt − t n,q,t f(t, s) − f(0, s) lim t = 0, t→0 t whenever the limit exists.

(iii) [2] We say that f(t, s) is uniformaly partialy differentiable at t = 0 ∈ I, with respect to s ∈ A ⊆ I if f(t, s) − f(0, s) lim t→0 t exists uniformly with respect to s ∈ A. From now on, we assume some appropriate conditions that imply the in- R ψ(t) tegrals of the form φ(t) g(t, s) dn,qs := F (t) exist, where g : I × I → R and φ, ψ : I → I. Lemma 3.2. The following statements are true (i) Assume that f(t, s) is continuous at t = 0 uniformly with respect to s ∈ [a]n,q. Then Z a F (t) = f(t, s) dn,qs 0 is continuous at t = 0. Z a (ii) Dn,q F (t) = Dn,q,t f(t, s) dn,qs, t 6= 0. 0

(iii) If Dn,q,t f(t, s) exists uniformly at t = 0 with respect to s ∈ [a]n,q, then Dn,q F (t) exists at t = 0 and Z a Dn,q F (0) = Dn,q,tf(0, s) dn,qs. 0

Proof. (i) Let  > 0. There exists δ > 0 such that

k k  ∀t ∈ I(|t| < δ =⇒ |f(t, q[k]n an ) − f(0, q[k]n an )| < , ∀k). |a| 2740 Alaa E. Hamza and M. H. Al-Ashwal

In view of corollary 2.3, for t ∈ I∩] − δ, δ[, we have ∞ X k k k h k k i [k]n n n n (n−1) [k]n n [k]n n F (t)− F (0) = − q a (q a − 1) f(t, q a )− f(0, q a ) k=0 < .

(ii) For t 6= 0, we have

∞ k k k [k]n n n n (n−1) X q a (q a − 1) k k D F (t) = − f(qtn, q[k]n an ) − f(t, q[k]n an ) n,q qtn − t k=0 ∞ k k k k X [k]n n n n (n−1) [k]n n = − q a (q a − 1) Dn,q,t f(t, q a ) k=0 Z a = Dn,q,t f(t, s) dn,qs. 0

(iii) Assume that Dn,q,t f(t, s) exists uniformly at t = 0 with respect to s ∈ [a]n,q. We conclude that

F (t) − F (0) Z a

− Dn,q,tf(0, s) dn,qs t 0 ∞ k k k [k]n n n n (n−1) X q a (q a − 1) k = − f(t, q[k]n an ) t k=0 ∞ k k k [k]n n n n (n−1) X q a (q a − 1) k + f(0, q[k]n an ) t k=0 ∞ X k k k k [k]n n n n (n−1) [k]n n + q a (q a − 1)Dn,q,t f(0, q a ) k=0 ∞ " k k [k]n n [k]n n X k k k f(t, q a ) − f(0, q a ) = − q[k]n an (qn an (n−1) − 1) t k=0 # k [k]n n − Dn,q,t f(0, q a ) .

For  > 0, there exists δ > 0 such that for all t ∈ I we have

k k [k]n n [k]n n f(t, q a ) − f(0, q a ) k  [k]n n 0 <|t|< δ =⇒ − Dn,q,t f(0, q a ) < . t |a| In view of corollary 2.3, for t ∈ I such that 0 < |t| < δ, we see that

F (t) − F (0) Z a

− Dn,q,tf(0, s) dn,qs < . t 0 n, q–Leibniz’s rule and n, q–Fubini’s theorem 2741

Corollary 3.3. If Dn,q,t f(t, s) exists uniformly at t = 0 with respect to s ∈ [a]n,q and s ∈ [b]n,q , then Z b Z b Dn,q,t f(t, s) dn,qs = Dn,q,t f(t, s) dn,qs, t ∈ I. a a Proof. By lemma 3.2, we get the desired result from the following equality Z b h Z b Z a i Dn,q,t f(t, s) dn,qs = Dn,q,t f(t, s) dn,qs − f(t, s) dn,qs . a 0 0

Theorem 3.4 (Leibniz’s integral rule). Define the function F by Z t F (t) := f(t, s) dn,qs. 0 The following statements are true (i) For t 6= 0, we have Z t n Dn,q F (t) = f(qt , t) + Dn,q,tf(t, s) dn,qs. (3) 0

(ii) If f(t, s) is continuous at (0, 0), then F is differentiable at t = 0 and F 0 (0) = f(0, 0). Proof. (i) At t 6= 0, we have

∞ k k k [k]n n n n n n (n−1) X q (qt ) (q (qt ) − 1) k D F (t) = − f(qtn, q[k]n (qtn)n ) n,q qtn − t k=0 ∞ k k k [k]n n n n (n−1) X q t (q t − 1)h k + f(t, q[k]n tn ) qtn − t k=0 k k i + f(qtn, q[k]n tn ) − f(qtn, q[k]n tn )

∞ k k k [k]n n n n n n (n−1) X q (qt ) (q (qt ) − 1) k = − f(qtn, q[k]n (qtn)n ) qtn − t k=0 ∞ k k k [k]n n n n (n−1) X q t (q t − 1) k + f(qtn, q[k]n tn ) qtn − t k=0 ∞ k k k [k]n n n n (n−1) X q t (q t − 1)h k k i − f(qtn, q[k]n tn ) − f(t, q[k]n tn ) qtn − t k=0 2742 Alaa E. Hamza and M. H. Al-Ashwal

" Z qtn Z t # 1 n n = n f(qt , s) dn,qs − f(qt , s) dn,qs qt − t 0 0 Z t f(qtn, s) − f(t, s) + n dn,qs 0 qt − t Z qtn Z t 1 n = n f(qt , s) dn,qs + Dn,q,tf(t, s) dn,qs. qt − t t 0 By theorem 2.10, we get Z t n Dn,q F (t) = f(qt , t) + Dn,q,t f(t, s) dn,qs. 0

(ii) Using corollary 2.3, we see that

F (s) − F (0)

− f(0, 0) s ∞ X k+1 k h k i [k+1]n n −1 [k]n n −1 [k]n n = − (q s − q s ) f(s, q s ) − f(0, 0) . k=0 The continuity of f(t, s) at (0, 0) implies that given  > 0, there exists k δ > 0 such that |s| < δ, ( and consequently q[k]n |s|n < δ ), implies k |f(s, q[k]n sn ) − f(0, 0)| < . Thus

F (s) − F (0)

− f(0, 0) < , s whenever 0 < |s| < δ, which completes the proof.

Theorem 3.5. Let φ : I → I be bounded. Define the function F by

Z φ(t) F (t) := f(t, s) dn,qs. 0 The following statements are true (i) The function F is n, q–differentiable at t 6= 0 and

Z φ(qtn) Z φ(t) 1 n Dn,q F (t) = n f(qt , s) dn,qs+ Dn,q,tf(t, s) dn,qs, t 6= 0. qt − t φ(t) 0 (4) n, q–Leibniz’s rule and n, q–Fubini’s theorem 2743

(ii) Assume that the following conditions hold

(1) φ is n, q–differentiable. (2) f(t, s) is uniformly partially differentiable at t = 0 with respect to s ∈ I such that Dn,q,tf(0, s) is continuous at s = 0. R t (3) The function H(t) = 0 f(0, s) dn,qs is differentiable at φ(0).

Then Dn,q F (t) exists at t = 0 and

φ(0) 0 Z Dn,q F (0) = f(0, φ(0)) φ (0) + Dn,q,tf(0, s) dn,qs. 0

Proof. (i) For t 6= 0, we have

∞ k k k [k]n n n n n n (n−1) Xq (φ(qt )) (q (φ(qt )) − 1) k D F (t) = − f(qtn, q[k]n (φ(qtn))n ) n,q qtn − t k=0 ∞ k k k [k]n n n n (n−1) X q (φ(t)) (q (φ(t)) − 1) k + f(qtn, q[k]n (φ(t))n ) qtn − t k=0 ∞ k k k [k]n n n n (n−1) X q (φ(t)) (q (φ(t)) − 1)h k − f(qtn, q[k]n (φ(t))n ) qtn − t k=0 k i − f(t, q[k]n (φ(t))n )

" Z φ(qtn) Z φ(t) # 1 n n = n f(qt , s) dn,qs − f(qt , s) dn,qs qt − t 0 0 Z φ(t) f(qtn, s) − f(t, s) + n dn,qs 0 qt − t Z φ(qtn) Z φ(t) 1 n = n f(qt , s) dn,qs + Dn,q,tf(t, s) dn,qs. qt − t φ(t) 0

(ii) To ensure the differentiability at t = 0, we write F as follows

F (t) = G(t) + (H ◦ φ)(t),

where

Z φ(t) G(t) = {f(t, s) − f(0, s)} dn,qs. 0 2744 Alaa E. Hamza and M. H. Al-Ashwal

R φ(0) First, we show that Dn,q G(0) = 0 Dn,q,tf(0, s) dn,qs. Indeed, one can see that

G(t) − G(0) Z φ(0)

− Dn,q,tf(0, s) dn,qs t 0

Z φ(t) hf(t, s) − f(0, s) i Z φ(t)

= − Dn,q,tf(0, s) dn,qs + Dn,q,tf(0, s) dn,qs 0 t 0

Z φ(0)

− Dn,q,tf(0, s) dn,qs 0

Z φ(t) hf(t, s) − f(0, s) i Z φ(t)

≤ − Dn,q,tf(0, s) dn,qs + Dn,q,tf(0, s) dn,qs 0 t 0

Z φ(0)

− Dn,q,tf(0, s) dn,qs . 0 Since f(t, s) is uniformly partially differentiable at t = 0 with respect to s ∈ I, and φ is bounded, then

Z φ(t) f(t, s) − f(0, s)

− Dn,q,tf(0, s) dn,qs → 0 as t → 0. 0 t R t The continuity of Dn,q,tf(0, s) at s=0 implies that K(t)= 0 Dn,q,tf(0, s) dn,qs is continuous at t = 0 [2] which in turn implies that K(φ(t)) is continuous at t = 0, that is Z φ(t) Z φ(0)

Dn,q,tf(0, s) dn,qs − Dn,q,tf(0, s) dn,qs → 0 as t → 0. 0 0 Consequently, we conclude that

G(t) − G(0) Z φ(0)

− Dn,q,tf(0, s) dn,qs → 0 as t → 0. (5) t 0 Since H is differentiable at φ(0) and φ is differentiable at 0, then H ◦φ is 0 0 0 differentiable at t = 0 and Dn,q(H◦φ)(0) = (H◦φ) (0) = H (φ(0))φ (0) = f(0, φ(0))φ(0). Therefore, we get the desired result.

Corollary 3.6. Let φ, ψ : I → I be bounded functions. Define the function F by Z ψ(t) F (t) := f(t, s) dn,qs. φ(t) Then, the following statements are true n, q–Leibniz’s rule and n, q–Fubini’s theorem 2745

(i) For t 6= 0, we have

" Z ψ(qtn) Z φ(qtn) # 1 n n Dn,q F (t) = n f(qt , s) dn,qs − f(qt , s) dn,qs qt − t ψ(t) φ(t) Z ψ(t) + Dn,q,t f(t, s) dn,qs, t 6= 0. φ(t)

(ii) Assume that the following conditions hold

(1) φ and ψ are n, q–differentiable. (2) f(t, s) is uniformly partially differentiable at t = 0 with respect to s ∈ I such that Dn,q,tf(0, s) is continuous at s = 0. R t (3) The function H(t) = 0 f(0, s) dn,qs is differentiable at φ(0) and ψ(0).

Then Dn,q F (t) exists at t = 0 and

ψ(0) 0 0 Z Dn,qF (0) = f(0, ψ(0)) ψ (0) − f(0, φ(0)) φ (0) + Dn,q,tf(0, s) dn,qs. φ(0)

Proof. By theorem 3.5 and using the definition

Z ψ(t) h Z ψ(t) Z φ(t) i Dn,q,t f(t, s) dn,qs = Dn,q,t f(t, s) dn,qs − f(t, s) dn,qs , φ(t) 0 0 we get the desired result.

Theorem 3.7 (Fubini’s theorem). Let f be defined on the closed rectangle R = [0, a]×[0, b] ⊂ I ×I. Assume that f(t, s) is continuous at t = 0 uniformly with respect to s ∈ [a]n,q and continuous at s = 0 uniformly with respect to t ∈ [b]n,q. Then, the double n, q–integrals

Z bZ a Z aZ b f(t, s) dn,qt dn,qs and f(t, s) dn,qs dn,qt 0 0 0 0 exist and they are equal, that is

Z bZ a Z aZ b f(t, s) dn,qt dn,qs = f(t, s) dn,qs dn,qt. 0 0 0 0 2746 Alaa E. Hamza and M. H. Al-Ashwal

R a R b Proof. By assumptions, lemma 3.2 tells us that 0 f(t, s) dn,qt and 0 f(t, s) dn,qs are continuous at s = 0 and at t = 0 respectively. Therefore both double n, q– integrals above exist and we have

b a b ∞ Z Z Z h j j j j i X [j]n n n n (n−1) [j]n n f(t, s) dn,qt dn,qs = q a (q a − 1) f(q a , s) dn,qs 0 0 0 j=0 ∞ ∞ X X k j k k j j j k = q[k]n+[j]n bn an (qn bn (n−1) − 1)(qn an (n−1) − 1) f(q[j]n an , q[k]n bn ) k=0 j=0 ∞ ∞ X X j k j j k k j k = q[j]n+[k]n an bn (qn an (n−1) − 1)(qn bn (n−1) − 1) f(q[j]n an , q[k]n bn ) j=0 k=0 a ∞ a b Z h k k k k i Z Z X [k]n n n n (n−1) [k]n n = q b (q b − 1) f(t, q b ) dn,qt = f(t, s) dn,qs dn,qt. 0 k=0 0 0

4 Conclusion

This article was devoted to prove Leibniz’s rule and Fubini’s theorem associ- ated with Power Quantum difference operators.There is a lot of work ahead of us. In one direction, one should ask about the n, q–Inequalities which de- pend on differentiation under the integral sign. Another direction, is to study in more details the Existence and uniqueness of solutions of Power Quantum difference equations, based on this operator.

Acknowledgements. We would like to express our gratitude to all mem- bers of the Seminar of the mathematical analysis in Dept. Math., Cairo Univ. for their valuable suggestions.

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Received: October 19, 2015; Published: December 3, 2015