A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus

: In this paper, we develop theorems on ﬁnite and inﬁnite summation formulas by utilizing the q and ( q , h ) anti-difference operators, and also we extend these core theorems to q ( α ) and ( q , h ) α difference operators. Several integer order theorems based on q and q ( α ) difference operator have been published, which gave us the idea to derive the fractional order anti-difference equations for q and q ( α ) difference operators. In order to develop the fractional order anti-difference equations for q and q ( α ) difference operators, we construct a function known as the quantum geometric and alpha-quantum geometric function, which behaves as the class of geometric series. We can use this function to convert an inﬁnite summation to a limited summation. Using this concept and by the gamma function, we derive the fractional order anti-difference equations for q and q ( α ) difference operators for polynomials, polynomial factorials, and logarithmic functions that provide solutions for symmetric difference operator. We provide appropriate examples to support our results. In addition, we extend these concepts to the ( q , h ) and ( q , h ) α difference operators, and we derive several integer and fractional order theorems that give solutions for the mixed symmetric difference operator. Finally, we plot the diagrams to analyze the ( q , h ) and ( q , h ) α difference operators for veriﬁcation.


Introduction
The study of calculus without limits is nowadays known as quantum calculus.Jackson's work [1] sheds light on the invention of q-calculus, often known as quantum calculus, while in 1908, Euler and Jacobi had already developed this type of calculus.The field of q-calculus emerged as a link between mathematics and physics.Numerous mathematical fields, including combinatorics, orthogonal polynomials, number theory, fundamental hyper-geometric functions, as well as other sciences, including mechanics, quantum theory, and the theory of relativity, make extensive use of it.
Most of the basic facts of quantum calculus are covered in the book by Kac and Cheung [2].Quantum calculus is a branch within the mathematical topic of time scales calculus.The q-differential equations are typically defined on a time scale set T q , where q is the scale index.Time scales offer a unifying framework for investigating the dynamic equations.The majority of the fundamental theory in the calculus of time scales was compiled in the text by Bohner and Peterson [3].
Though quantum calculus plays a major role in physics, engineers and mathematics also show interest in fractional q-difference equations and q-calculus.The main focus of developing q-difference equations is to characterize some unique physical processes and other areas.Some of the topics that have been developed and investigated in conjunction with the creation of the q-calculus theory include the q-Gamma, q-Laplace transform, q-Taylor expansion, q-Beta functions, q-integral transforms theory, q-Mittag Leffler functions, and others.Refer to the articles [4][5][6][7][8][9][10][11][12] for additional information on fractional and qcalculus equations with q-differentials.The study of fractional q-calculus is still in its early stages when compared to the classical fractional calculus.
In recent years, there has been some research on the uniqueness and existence of solutions to fractional q-calculus.In [13], the authors suggested a technique for solving several linear fractional q-differential equations that involves corresponding integer order equations.Abdeljawad et al. demonstrated the uniqueness of a nonlinear delay Caputo fractional q-difference system initial value problem in [14] by employing a new extended form of the discrete fractional q-Gronwall inequality, whereas the author provided the applications in [15,16].By utilizing the Banach's contraction mapping concept and by using the p-Laplacian operator, the authors of [17] showed that the Caputo q-fractional boundary value problem has a unique solution.[18] The contraction mapping principle was used by Ren et al.They also used traditional fixed point theorems to prove that numerous positive solutions exist under certain conditions.In [19], Zhang et al. provided the uniqueness and existence of solutions to the Caputo fractional q-differential equations, and also in [20] they considered the possibility of a singular solution in the q-integral space.The authors in [21] provided the applications of quantum calculus to impulsive difference equations on finite intervals.The applications of the q-calculus to the problem of a falling body in a resisting medium have been given in [22].Later, the authors in [23] developed the q-symmetric derivative, which is defined as (u(qk) − u(q −1 k))/(qk − q −1 k).
The q-differential operator is then extended to q-difference operator.The q-difference operator was proposed by the authors [24] in 2014 and is defined as d q u(k) = u(kq) − u(k), and the oscillation of q difference equation was discussed in [25].In [26], the authors suggested the d q (α) operator by defining d q (α) u(k) = u(kq) − αu(k).In 2022, the authors in [27] developed the q-symmetric difference operator, that is D q u(k) = u(kq) − u(kq −1 ), which is the combination of forward and backward q-difference operator.Here, the authors developed the theorems for integer order using the q-difference operator that generates a solution for the q-symmetric difference operator.This motivates us to develop the fractional order theorems for q-symmetric difference operator.In addition, we have extended this q-symmetric difference operator to (q, h)-symmetric operator which is defined as , and its alpha (q, h)-symmetric operator is defined as Throughout this paper, we concentrate only on the development of fractional order q and (q, h) anti-difference equations, and we have extended these core theorems to q (α) and (q, h) α fractional anti-difference equations.Those findings will provide fractional order solution for the (q, h) and (q, h) α symmetric difference operator.Here, the findings are based only on the delta operator.One can do the same for the nabla operator.This is how the paper is structured.The Introduction is the focus of Section 1.In Section 2, we discuss the preliminaries of q and q (α) difference operator.In Sections 3-5, we develop the integer and fractional order theorems for q, q (α) , (q, h), and (q, h) α difference operators.The conclusion is covered in Section 6.

Definition 1 ([25]
).Let u : T q → R and 1 = q > 0 ∈ R. The q and q (α) difference operator (q and q (α) -symmetric difference operator), denoted as d q and d q (α) , on u(k) are, respectively, defined as (2) Definition 2 ([25]).If there exists a function v : T q → R such that d q u(k) = v(k), then its inverse q and q (α) difference operator, denoted as I q and I q (α) are, respectively, defined as where c is a constant.

Definition 3 ([27]
).Let n ∈ N and k, q be any real number.Then, the q-polynomial falling factorial function of k q is defined as Lemma 1 ([27]).The power rule for q and q (α) difference operator is as follows: 1.
If n ∈ N and q = 1, then for k ∈ R, q and I q k 2.
If n ∈ N, q n =α and α∈ R, then for k ∈ R, q and I q (α) k Lemma 2 ([27]).Let u, v : T q → R and q = 1 ∈ R. The product rule of q and q (α) difference operator is, respectively, defined by and Result 1 ([27]).Let k, α∈ R and q ∈ R −{0, 1}.Then, I q (1) = log(k)/ log(q) and I q Result 2 ([27]).If 1 = q > 0 and α< 1 ∈ R, then for k ∈ (0, ∞), we have and 3. Fundamental Theorems for q and q (α) Symmetric Difference Operator In this section, we present some basic notions of polynomial factorial function and gamma function.Then, we use the q and q (α) difference operator and its inverse operators to derive fundamental theorems.Definition 4 ([28]).Let k ∈ R and n ∈ N.Then, the falling factorial function is defined as For 0 < ν∈ R and k ∈ R, the generalized gamma function is where k + 1 and k − ν + 1 is non-equal to zero or a negative integer.
Lemma 3 ([29]).For the first n natural numbers, the x th power polynomial factorial is

Fundamental Theorems for q Operator
Using the q symmetric difference operator, we develop a few theorems for integer order (x-th order) and fractional order (ν-th order) sums.Definition 5. Let s, k ∈ R, q ∈ R −{0, 1} such that s ∈ T q and u : T q → R be a function.Then, the quantum geometric function (or q-geometric function) is defined as Proof.The proof completes by replacing k by k + 1 and r by r + 1 in Definition 5.
Lemma 5.If x ∈ N and assuming the conditions given in Lemma 4, then Proof.Equation ( 17) can be represented as From (19), one can easily find the next term as ) .
Similarly, we can find Hence, the proof completes by replacing x by x + 1 in Equation (20).
Then, the anti-difference principle of q difference operator is given by Proof.Since When k is substituted for (k/q) in Equation ( 22), we obtain Once again, by changing k to (k/q) in Equation ( 23), we obtain Substituting Equation (24) in Equation ( 23), we obtain Proceeding like this up to n times, we obtain Applying lim n→∞ in the previous equation and assuming v(0) = u(0) = 0, we obtain Replacing k by s and r by k in (26), we obtain From Equation (3), we arrive at Finally, the proof completes by substituting Equation ( 17) in the previous equation.
Thus, for the q difference operator, the higher order anti-difference principle is given by Proof.Theorem 1 provides the proof for x = 1.
If we apply the I q operator on both sides of Equation ( 26), we obtain I 2 q u(k) = I q u(k/q) + I q u(k/q 2 ) + I q u(k/q 3 ) + I q u(k/q 4 ) + I q u(k/q 5 ) + . . .
Replacing the right side of the aforementioned equation by (26), we obtain which implies Replacing k by s and r by k in Equation ( 29), we arrive at Once again, by using the I q operator on both sides of the expression (29), we obtain Inserting Equation (26) in each term of the right side of the previous equation, we obtain The above equation will be written as By Lemma 3 for m = 1, the above equation becomes Replacing k by s and r by k in Equation ( 30), we obtain Similarly, the fourth inverse will be Following the similar manner, we obtain the general term as Hence, by Lemma 5, we obtain (28).
Similarly, applying the I q operator on the function k (2) q for x times, we obtain (2) q /(q 2 − 1) x .
The following Definition 6 is the generalized version for Definition 5.
(Γ(r + ν)/Γ(r + 1))u(s/q r+ν ) be convergent such that s ∈ T q and u : T q → R be a function.Then, for ν > 0, the generalized quantum geometric function (or generalized q-geometric function) is defined as where The following Theorem 3 is the generalized version for Theorem 2.
Example 2. Taking u(k) = k 2 and ν = 2.7 in Equation (36), we obtain Using Equation (1) and then applying the I q operator on the function k 2 q for x times, we obtain For any real ν > 0, Equation (39) becomes Taking s = 8.1, k = 6 and q = 3.2 in Equation (38), we arrive The right side of Equation (38) becomes Hence, substituting Equations (40) and (41) in Equation (38), we obtain the result.

Fundamental Theorems for q (α) Operator
By utilizing the q (α) symmetric difference operator, we developed theorems for integer order (or m-th order) and the fractional order (ν-th order).Here, the difference operator q (α) changes to the q-difference operator if α = 1.
α r u(s/q r+j ) be convergent such that s ∈ T q and u : T q → R is a function.Then the alpha-quantum geometric function (or q (α)geometric function) is defined as Lemma 6.Consider the conditions given in Definition 7. If x ∈ N, then Proof.Equation (42) can be written as From (44), one can easily find the next term as ) .
Similarly, we can find Hence, the proof completes by replacing x by x + 1 in Equation (45).
The following Definition 8 is the generalized version for Definition 7.
α r u(s/q r+ν ) is convergent such that s ∈ T q and u : T q → R be a function.Then, for ν > 0, the generalized quantum geometric function (or generalized q-geometric function) is defined as where The following Theorem 6 is the generalized version for Theorem 5.

Theorem 6. (Generalized
, Then, the ν-th order (real order) anti-difference principle of q (α) difference operator is given by Proof.The proof is similar to Theorem 3 using Equations ( 4) and (53).
The integer and fractional order anti-difference equations developed in this section provides the solution for q and q (α) symmetric difference operators.

Mixed Symmetric Difference Operator
In this section, we derive some fundamental theorems using (q, h) difference operator and its inverse operators.Here, we introduce the infinite set M q h = {k, kq + h, kq 2 + 2h, . ..} satisfying the condition that for any k ∈M q h implies kq ±1 ± h ∈M q h for any fixed number 0 = k ∈ R. One can refer the h-difference operator in [30].Definition 9. Let u :M q h → R be a function.Then, the (q, h) difference operator (mixed symmetric difference operator), denoted by ∆ (q,h) is defined as Definition 10.Let h, q, k ∈ R and n ∈ N. The (q, h) polynomial factorial function k Then, the product rule of (q, h) difference operator is obtained as Proof.Applying the operator ∆ (q,h) on the function u(k)v(k) and then adding and subtracting the term u(k)w(kq + h), we obtain Thus, the proof completes by taking ∆ (q,h) Property 1.Some of the properties of (q, h) difference operator are given below: (i) If q = 1, then (60) becomes h-difference operator.
(iv) The solution does not exist if we take q = 1 and h = 0 simultaneously.

Integer Order Theorems
Here, we develop several theorems for integer order (x-th order) using the (q, h) difference operator.
Then, the antidifference principle of (q, h) operator is given by From Definition 9, Equation (64) becomes The above equation can be represented as Replacing k by k/q in Equation (65), we obtain v q(k/q) + h = u(k/q) + v(k/q) which implies Replacing Replacing k by (k − h)/q in Equation (67), we arrive at The aforementioned equation can be written as Now, substituting Equation (68) in Equation (67), we obtain Again, replacing k by (k − h − qh)/q 2 in Equation (67), we obtain which is the same as Substituting Equation (70) in Equation (69), we obtain Similarly, replacing k by (k − h 2 ∑ r=0 q r )/q 3 in Equation (67), we obtain Substituting Equation (72) in Equation (71), we obtain Similarly, again replacing k by (k − h(q 3 + q 2 + q + 1))/q 4 in Equation ( 67), and then substituting Equation (67) in Equation ( 73), we arrive at Proceeding in a similar manner for n times, we obtain the general term as Remark 1.The operators are the first order q and h difference operators, respectively.
That is, which completes the proof.
Then, the m-th order of (q, h) difference equation is given by Proof.The proof completes by replacing in Equation (78).

Mixed Alpha Symmetric Difference Operator
In this section, we develop fundamental theorems using (q, h) α difference operator and its inverse operators.If we take α = 1, then the (q, h) α difference equation will become (q, h) difference equation.
Lemma 8.If u, v :M q h → R, q ∈ R −{0, 1}, 0 = h ∈ R and α∈ R.Then, the product rule of (q, h) α difference operator is obtained as (100) Proof.The proof is similar to Lemma 7 by using the ∆ (q,h) α operator.
(iv) The solution does not exist if we take q = 1, h = 0 and α = 1 simultaneously.